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This exam covers

laws of indices
using surds and rationalising the denominator
expanding brackets
simplifying expressions
solving linear inequalities
finding common factors
dividing a polynomial with remainders, using algebraic division
factor theorem
remainder theorem
inverse and composite functions

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This question aims to test understanding and ability to use the laws of indices.

", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

Using the laws of indices, simplify each expression down to its simplest form. Recall that $a^{0} = 1$ for any number $a$.

", "advice": "

a)

Here we are using the rule of indices: $a^m \\times a^n = a^{m+n}$.

Using this rule,

\\[
\\begin{align}
a^\\var{x} \\times a^\\var{y}\\ &= a^\\simplify[all, !collectNumbers]{{x}+{y}}\\\\
&= a^\\var{x+y}.
\\end{align}
\\]

b)

We are asked to find $\\var{c}a^\\var{p} \\times \\var{d}a^\\var{q}$.

Notice there is a constant in front of each of the terms.

To do this, write the product out explicitly, as

\\[\\var{c}a^\\var{p} \\times \\var{d}a^\\var{q} = \\var{c} \\times \\var{d} \\times a^\\var{p} \\times a^\\var{q}.\\]

We know that $\\var{c} \\times \\var{d} = \\var{c*d}$, and using the rule of indices: $a^\\var{p} \\times a^\\var{q} = a^\\var{p+q}$.

Therefore:

\\begin{align}
\\var{c}a^\\var{p} \\times \\var{d}a^\\var{q}&= \\var{c*d} \\times a^\\var{p+q} \\\\
&= \\simplify{{c*d}*a^{p+q}}.
\\end{align}

c)

Here we are using: $a^m \\div a^n = a^{m-n}$.

We are asked to simplify the expression, $\\displaystyle\\simplify{{b}*a^{x}/({g}*a^{y})}$.

To do this, we just have to use the previously mentioned rule of indices. We write this out explicity as

\\[\\simplify{{b}*a^{x}/({g}*a^{y})} = \\simplify{{b}/{g}} \\times \\simplify{a^{x}/(a^{y})}.\\]

Using rules of indices,

\\begin{align} \\frac{a^\\var{x}}{a^\\var{y}} &= a^\\var{x} \\div a^\\var{y}\\\\
&= a^\\simplify[all, !collectNumbers]{{x}-{y}}\\\\
&= a^\\var{x-y}.
\\end{align}

Therefore,

\\begin{align}
\\frac{\\var{b}a^\\var{x}}{\\var{g}a^\\var{y}} &= \\simplify{{b}/{g}} \\times \\simplify{a^{{x}-{y}}}\\\\
&= \\simplify{{b}/{g}*a^{x-y}}.
\\end{align}

Alternatively,

Using the rule of indices: $a^{-m} = \\displaystyle\\frac{1}{a^{m}}$, we can rewrite the question as:

\\begin{align}
\\frac{\\var{b}a^\\var{x}}{\\var{g}a^\\var{y}} &= \\simplify{{b}/{g}} \\times \\frac{a^\\var{x}}{a^\\var{y}}\\\\
&= \\simplify{{b}/{g}} \\times a^\\var{x} \\times a^{-\\var{y}}.
\\end{align}

And then using the rule: $a^m \\times a^n = a^{m+n}$, this becomes:

\\begin{align}
\\simplify{{b}/{g}} \\times a^\\var{x} \\times a^{-\\var{y}} &= \\simplify{{b}/{g}} \\times a^\\simplify[all,!collectNumbers]{{x}+(-{y})}\\\\
&= \\simplify{{b}/{g}*a^{x-y}}.
\\end{align}

d)

The question asks us to simplify $(\\simplify{{c}*a^{p}})^{\\var{q}}$.

To do this we use the rules:

\\[(a^{m})^{n} = a^{mn},\\]

\\[(ab)^m = a^mb^m.\\]

We can then expand the equation as

\\[(\\simplify{{c}*a^{p}})^{\\var{q}}= \\var{c}^{\\var{q}} \\times (a^{\\var{p}})^{\\var{q}}.\\]

Then using the rule of indices mentioned previously,

\\[
\\begin{align}
(\\simplify{{c}*a^{p}})^{\\var{q}}&= \\simplify{{c}^{q}} \\times a^\\var{p*q}\\\\
&= \\simplify{{c}^{q}*a^{p*q}}.
\\end{align}
\\]

e)

The question asks us to simplify $\\sqrt[\\var{d}]{\\var{x}^\\var{d}a}$.

To do this we use the rules:

\\[a^\\frac{1}{m} = \\sqrt[m]{a},\\]

\\[(ab)^m = a^mb^m.\\]

We can expand the expression as follows:

\\[
\\begin{align}
\\sqrt[\\var{d}]{a} &= (\\simplify{a})^\\frac{1}{\\var{d}}\\\\
&= a^\\frac{1}{\\var{d}}.
\\end{align}
\\]

f)

The question requires us to simplify $\\sqrt[\\var{c}]{a^\\var{q}}$.

Here, we use the rule of indices: $a^\\frac{n}{m} = \\sqrt[m]{a^n}$, allowing us to expand the expression as follows:

\\[
\\begin{align}
\\sqrt[\\var{c}]{\\simplify{a^{q}}} &= \\simplify[fractionnumbers,all]{(a^{q})^{{1}/{{c}}}}\\\\
&= \\simplify[fractionnumbers,all]{a^{{q}/{c}}}.
\\end{align}
\\]

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Used in parts a,c and f

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Write $a^{\\var{x}} \\times a^{\\var{y}}$ as a single power of $a$.

$a^{\\var{x}} \\times a^{\\var{y}} =$ [[0]].

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Use the rule: $a^m \\times a^n = a^{m+n}$.

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Write $\\var{c}a^\\var{p} \\times \\var{d}a^\\var{q}$ as an integer multiplied by a single power of $a$.

$\\var{c}a^\\var{p} \\times \\var{d}a^\\var{q} =$ [[0]].

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Write $\\displaystyle\\simplify{{b}*a^{x}/({g}*a^{y})}$ as a number multiplied by a single power of $a$.

$\\displaystyle\\simplify{{b}*a^{x}/({g}*a^{y})} =$ [[0]].

You could use one of the following rules:

$a^m \\div a^n = a^{m-n}$.

$a^{-m} = \\displaystyle\\frac{1}{a^m}$.

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Write $(\\simplify{{c}*a^{p}})^{\\var{q}}$ as an integer multiplied by a single power of $a$.

$(\\simplify{{c}*a^{p}})^{\\var{q}} =$ [[0]].

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Use the rules:

$(ab)^m = a^mb^m$.

$(a^m)^n = a^{mn}$.

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Write $\\sqrt[\\var{d}]{a}$ as a single power of $a$.

$\\sqrt[\\var{d}]{a} =$ [[0]].

Use the rule: $a^\\frac{1}{m} = \\sqrt[m]{a}$.

You must input your answer as a single power of a.

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Write $\\sqrt[\\var{q}]{a^\\var{c}}$ as a single power of $a$.

$\\sqrt[\\var{q}]{a^\\var{c}} =$ [[0]].

Use the rule: $a^\\frac{n}{m} = \\sqrt[m]{a^n}$.

You must input your answer as a single power of a.

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Manipulate surds and rationalise the denominator of a fraction when it is a surd.

", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

To include a square root sign in your answer use sqrt(). For example, to write $\\sqrt{3}$, type sqrt(3) into the answer box. If you are entering a number multiplied by the square root of some other number, for example $3\\sqrt{5}$, type 3*sqrt(5) into the answer box.

", "advice": "

a)

Surds can be manipulated using the rule

\\[\\sqrt{(ab)} = \\sqrt{a} \\times \\sqrt{b}.\\]

We are asked to state which of $\\sqrt{\\var{p}}$, $\\sqrt{\\simplify{{a}*{n}^2}}$, and $\\sqrt{\\var{a}}$ can be simplified further. Commonly, surds can be simplified if the number inside of the square root has a square number as a factor.

Here, $\\var{p}$ is a prime number which means that its only divisors are $\\var{p}$ and $1$.

Therefore, $\\sqrt{\\var{p}}$ cannot be simplified any further.

Similarly, $\\var{a}$ is also a prime number, so $\\sqrt{\\var{a}}$ also cannot be simplified any further.

On the other hand, $\\simplify{{a}*{n}^2}$ is not a prime number and we can use the previous rule to simplify $\\sqrt{\\simplify{{a}*{n}^2}}$ as

\\[
\\begin{align}
\\sqrt{\\simplify{{a}*{n}^2}} &= \\sqrt{\\simplify{{n}^2}} \\times \\sqrt{\\var{a}}\\\\
&= \\simplify{{n}*sqrt({a})}.
\\end{align}
\\]

b)

Using the same rule of manipulation as in part a), we can simplify $\\sqrt{\\simplify{{n}^2*{p}}}$ as

\\[
\\begin{align}
\\sqrt{\\simplify{{n}^2*{p}}} &= \\sqrt{\\simplify{{n}^2}} \\times \\sqrt{\\var{p}}\\\\
&= \\simplify{{n}*sqrt({p})}.
\\end{align}
\\]

c)

Here, we can use both of the rules for manipulating surds:

\\[\\sqrt{(ab)} = \\sqrt{a} \\times \\sqrt{b} \\text{.} \\]

\\[ \\sqrt{\\frac{a}{b}} = \\frac{\\sqrt{a}}{\\sqrt{b}} \\text{.} \\]

We can simplify $\\displaystyle\\frac{ \\sqrt{\\simplify{{a}*{v}}} }{ \\sqrt{\\var{a}} }$ as follows.

\\[
\\begin{align}
\\frac{\\sqrt{\\simplify{{a}*{v}}}}{\\sqrt{\\var{a}}} &= \\frac{\\sqrt{\\var{a}} \\times \\sqrt{\\var{v}}}{\\sqrt{\\var{a}}} \\\\[0.5em]
&= \\frac{\\sqrt{\\var{a}}}{\\sqrt{\\var{a}}} \\times \\sqrt{\\var{v}} \\\\[0.5em]
&= \\simplify{{sqrt(a)/sqrt(a)}} \\times \\sqrt{\\var{v}} \\\\[0.5em]
&= \\sqrt{\\var{v}} \\text{.}
\\end{align}
\\]

Or,

\\[
\\begin{align}
\\frac{\\sqrt{\\simplify{{a}*{v}}}}{\\sqrt{\\var{a}}} &= \\sqrt{\\frac{\\simplify{{a}*{v}}}{\\var{a}}} \\\\[0.5em]
&= \\sqrt{\\var{v}} \\text{.}
\\end{align}
\\]

d)

We can simplify the fraction as

\\[
\\begin{align}
\\frac{\\sqrt{\\simplify{({b}{m})^2*{s}}}}{\\var{m}} &= \\frac{\\sqrt{\\simplify{({b*m})^2}} \\times \\sqrt{\\var{s}}}{\\var{m}} \\\\[0.5em]
&= \\frac{\\simplify{{b*m}} \\times \\sqrt{\\var{s}}}{\\var{m}} \\\\[0.5em]
&= \\simplify{{b}*sqrt({s})} \\text{.}
\\end{align}
\\]

e)

\\[
\\begin{align}
\\simplify{{d}sqrt({a}) - {b}sqrt({v}^2{a})+{n}sqrt({b}^2*{a})} &= \\var{d}\\sqrt{\\var{a}} - \\var{b}(\\sqrt{\\simplify{{v}^2}} \\times \\sqrt{\\var{a}})+\\var{n}(\\sqrt{\\simplify{{b}^2}} \\times \\sqrt{\\var{a}}) \\\\
&= \\var{d}\\sqrt{\\var{a}} -\\var{b}(\\simplify{{v}*sqrt({a})})+\\var{n}(\\simplify{{b}*sqrt({a})}) \\\\
&= \\simplify{{d}sqrt({a})}-\\simplify{{b}*{v}sqrt({a})}+\\simplify{{n}*{b}sqrt({a})} \\\\
&= \\simplify{({d}-{b}*{v}+{n}*{b})sqrt({a})} \\text{.}
\\end{align}
\\]

f)

We rationalise the denominator of fractions of the form $\\displaystyle\\frac{1}{\\sqrt{a}}$, by multiplying the top and bottom by $\\sqrt{a}$.

Therefore, to rationalise the denominator of the fraction $\\displaystyle\\frac{1}{\\sqrt{\\var{a}}}$, we multiply top and bottom by $\\sqrt{\\var{a}}$.

\\[
\\begin{align}
\\frac{1}{\\sqrt{\\var{a}}} &= \\frac{1}{\\sqrt{\\var{a}}} \\times \\frac{\\sqrt{\\var{a}}}{\\sqrt{\\var{a}}} \\\\[0.5em]
&= \\frac{\\sqrt{\\var{a}}}{\\var{a}} \\text{.}
\\end{align}
\\]

g)

We rationalise the denominator of fractions of the form $\\displaystyle\\frac{1}{a+\\sqrt{b}}$ by multiplying the top and bottom by $a-\\sqrt{b}$.

Therefore, to rationalise the denominator of the fraction $\\displaystyle\\frac{1}{\\var{n}+\\sqrt{\\var{a}}}$, we multiply the top and bottom by $\\var{n} - \\sqrt{\\var{a}}$.

\\[
\\begin{align}
\\frac{1}{\\var{n}+\\sqrt{\\var{a}}} &= \\frac{1}{\\var{n}+\\sqrt{\\var{a}}} \\times \\frac{\\var{n}-\\sqrt{\\var{a}}}{\\var{n}-\\sqrt{\\var{a}}} \\\\[0.5em]
&=\\frac{\\var{n}-\\sqrt{\\var{a}}}{(\\var{n}+\\sqrt{\\var{a}})(\\var{n}-\\sqrt{\\var{a}})} \\\\[0.5em]
&=\\frac{\\var{n}-\\sqrt{\\var{a}}}{\\simplify{{n}^2}-\\var{a}} \\\\[0.5em]
&=\\frac{\\var{n}-\\sqrt{\\var{a}}}{\\simplify{{n}^2-{a}}} \\text{.}
\\end{align}
\\]

h)

We rationalise the denominator of fractions of the form $\\displaystyle\\frac{1}{a-\\sqrt{b}}$ by multiplying the top and bottom by $a+\\sqrt{b}$.

Therefore, to rationalise the denominator of the fraction $\\displaystyle\\frac{\\var{t}}{\\var{d+p}-\\sqrt{\\var{p}}}$, we multiply the top and bottom by $\\var{d+p}+\\sqrt{\\var{p}}$.

\\[
\\begin{align}
\\frac{\\var{t}}{\\var{d+p}-\\sqrt{\\var{p}}} &= \\frac{\\var{t}}{\\var{d+p}-\\sqrt{\\var{p}}} \\times \\frac{\\var{d+p}+\\sqrt{\\var{p}}}{\\var{d+p}+\\sqrt{\\var{p}}} \\\\[0.5em]
&=\\frac{\\var{t}(\\var{d+p}+\\sqrt{\\var{p}})}{(\\var{d+p}-\\sqrt{\\var{p}})(\\var{d+p}+\\sqrt{\\var{p}})} \\\\[0.5em]
&=\\frac{\\var{t}(\\var{d+p}+\\sqrt{\\var{p}})}{\\simplify{{d+p}^2}-\\var{p}} \\\\[0.5em]
&=\\frac{\\var{t}(\\var{d+p}+\\sqrt{\\var{p}})}{\\simplify{{d+p}^2-{p}}} \\\\[0.5em]
&=\\simplify{{t}/{(d+p)^2-p}}(\\var{d+p}+\\sqrt{\\var{p}}) \\\\[0.5em]
&= \\simplify[all,!noleadingMinus]{({t*(d+p)}+{t}*sqrt({p}))/({(d+p)^2-p})} \\text{.}
\\end{align}
\\]

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shorter list of primes for parts a,c,e,f and g

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Which of the following can be simplified further?

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Can be simplified further

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Cannot be simplified further

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$\\sqrt{\\var{p}}$

", "

$\\sqrt{\\simplify{{a}*{n}^2}}$

", "

$\\sqrt{\\var{a}}$

"]}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Simplify $\\sqrt{\\simplify{{n}^2*{p}}}$.

$\\sqrt{\\simplify{{n}^2*{p}}} =$ [[0]]$\\sqrt{\\var{p}}$.

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Recall the first rule of surds

$\\sqrt{(ab)} = \\sqrt{a} \\times \\sqrt{b}$.

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Simplify $\\displaystyle\\frac{\\sqrt{\\simplify{{a}*{v}}}}{\\sqrt{\\var{a}}}$.

$\\displaystyle\\frac{\\sqrt{\\simplify{{a}*{v}}}}{\\sqrt{\\var{a}}} =$ [[0]].

You could use either of the following rules:

$\\sqrt{(ab)} = \\sqrt{a} \\times \\sqrt{b}$.

$\\displaystyle\\sqrt{\\frac{a}{b}} = \\displaystyle\\frac{\\sqrt{a}}{\\sqrt{b}}$.

You must simplify your answer further.

"}, "notallowed": {"strings": ["/"], "showStrings": false, "partialCredit": 0, "message": "

You must simplify your answer further.

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Simplify $\\displaystyle\\frac{\\sqrt{\\simplify{({b}{m})^2*{s}}}}{\\var{m}}$.

$\\displaystyle\\frac{\\sqrt{\\simplify{({b}*{m})^2*{s}}}}{\\var{m}} =$ [[0]]$\\sqrt{\\var{s}}$.

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Simplify $\\simplify{{d}sqrt({a}) - {b}sqrt({v}^2*{a})+{n}sqrt({b}^2*{a})}$.

$\\simplify{{d}sqrt({a}) - {b}sqrt({v}^2*{a})+{n}sqrt({b}^2*{a})} =$ [[0]].

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Rationalise the denominator of the fraction $\\displaystyle\\frac{1}{\\sqrt{\\var{a}}}$.

$\\displaystyle\\frac{1}{\\sqrt{\\var{a}}} =$ [[0]] [[1]] .

To rationalise the denominator of fractions in the form $\\frac{1}{\\sqrt{a}}$, multiply the top and bottom by $\\sqrt{a}$.

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Rationalise the denominator of the fraction $\\displaystyle\\frac{1}{\\var{n}+\\sqrt{\\var{a}}}$.

$\\displaystyle\\frac{1}{\\var{n}+\\sqrt{\\var{a}}} =$ [[0]] [[1]] .

To rationalise the denominator of fractions in the form $\\displaystyle\\frac{1}{a+\\sqrt{b}}$, multiply the top and bottom by $a-\\sqrt{b}$.

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Rationalise the denominator of the fraction $\\displaystyle\\frac{\\var{t}}{\\var{d+p}-\\sqrt{\\var{p}}}$.

$\\displaystyle\\frac{\\var{t}}{\\var{d+p}-\\sqrt{\\var{p}}} =$ [[0]] [[1]] .

To rationalise the denominator of fractions in the form, $\\displaystyle\\frac{1}{a-\\sqrt{b}}$, multiply the top and bottom by ${a+\\sqrt{b}}$.

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This question is made up of 10 exercises to practice the multiplication of brackets by a single term.

", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

Expand the expressions below by multiplying each of the terms inside the brackets by the term outside. Give each answer in its simplest form.

", "advice": "

Expand brackets using the general formula $\\displaystyle a(x+c)=ax+ac$. This means we multiply each term inside the brackets by the term outside the brackets.

It is easy to forget that the sign outside the brackets also needs to be involved in the multiplication so remember that when two of the same sign are multiplied, the resultant term is positive and when opposite signs are multiplied, the result is negative.

a)

\\[
\\begin{align}
\\simplify[terms]{{a[1]}({a[2]}x+{a[3]})}&=
\\simplify[!collectNumbers]{({a[1]}{a[2]})x+({a[1]}{a[3]})}\\\\&
=\\simplify{{a[1]}*{a[2]}x+{a[1]}{a[3]}}\\text{.}
\\end{align}
\\]

b)

\\[
\\begin{align}
\\simplify[terms]{{a[4]}({a[5]}x+{a[6]})}&=
\\simplify[!collectNumbers]{{a[4]}{a[5]}x+{a[4]}{a[6]}}\\\\&=
\\simplify{{a[4]}*{a[5]}x+{a[4]}{a[6]}}\\text{.}
\\end{align}
\\]

c)

\\[
\\begin{align}
\\simplify[terms]{{a[7]}({a[8]}x^2+{a[9]}y)}&=
\\simplify[!collectNumbers]{{a[7]}{a[8]}x^2+{a[7]}{a[9]}y}\\\\&=
\\simplify{{a[7]}*{a[8]}x^2+{a[7]}*{a[9]}y}\\text{.}
\\end{align}
\\]

d)

\\[
\\begin{align}
\\simplify[terms]{{a[10]}({a[11]}x^2+{a[12]}y)}&=
\\simplify[!collectNumbers]{{a[10]}{a[11]}x^2+{a[10]}{a[12]}y}\\\\&=
\\simplify{{a[10]}*{a[11]}x^2+{a[10]}*{a[12]}y}\\text{.}
\\end{align}
\\]

e)

\\[
\\begin{align}
\\simplify[terms]{{a[13]}x({a[14]}x^2+{a[15]}x+{a[16]})}&=
\\simplify[!collectNumbers]{{a[13]}x{a[14]}x^2+{a[13]}x{a[15]}x+{a[13]}x{a[16]}}\\\\&=
\\simplify{{a[13]}{a[14]}x^3+{a[13]}{a[15]}x^2+{a[13]}{a[16]}x}\\text{.}
\\end{align}
\\]

f)

\\[
\\begin{align}
\\simplify[terms]{{a[17]}x({a[18]}x^2+{a[19]}x+{a[20]})}&=
\\simplify[!collectNumbers]{{a[17]}x{a[18]}x^2+{a[17]}x{a[19]}x+{a[17]}x{a[20]}}\\\\&=
\\simplify{{a[17]}{a[18]}x^3+{a[17]}{a[19]}x^2+{a[17]}{a[20]}x}\\text{.}
\\end{align}
\\]

g)

\\[
\\begin{align}
\\simplify[terms]{{a[21]}x({a[22]}x^2+{a[23]}x)+{a[24]}x^2+{a[25]}x^3}&=
\\simplify[!collectNumbers]{x^2({a[21]}{a[23]})+x^2{a[24]}+x^3({a[21]}{a[22]})+x^3{a[25]}}\\\\&=
\\simplify[!collectNumbers]{x^2({a[21]}{a[23]}+{a[24]})+x^3({a[21]}{a[22]}+{a[25]})}\\\\&=
\\simplify{x^2({a[21]}{a[23]}+{a[24]})+x^3({a[21]}{a[22]}+{a[25]})}\\text{.}
\\end{align}
\\]

h)

\\[
\\begin{align}
\\simplify[terms]{({a[26]}x^2+{a[27]}x^3)+{a[28]}x({a[29]}x^2+{a[30]}x)}&=
\\simplify[!collectNumbers]{x^2({a[26]})+x^2({a[28]}{a[30]})+x^3({a[28]}{a[29]})+x^3({a[27]})}\\\\&=
\\simplify[!collectNumbers]{x^2({a[26]}+{a[28]}{a[30]})+x^3({a[28]}{a[29]}+{a[27]})}\\\\&=
\\simplify{x^2({a[26]}+{a[28]}{a[30]})+x^3({a[28]}{a[29]}+{a[27]})}\\text{.}
\\end{align}
\\]

i)

\\[
\\begin{align}
\\simplify[terms]{{a[31]}({a[32]}x+{a[33]}y)+{a[34]}x({a[42]}+{a[35]}y)}&=
\\simplify[!collectNumbers]{({a[31]}{a[32]})x+({a[34]}{a[42]})x+{a[31]}{a[33]}y+{a[34]}{a[35]}x*y}\\\\&=
\\simplify[!collectNumbers]{({a[31]}{a[32]}+{a[34]}{a[42]})x+{a[31]}{a[33]}y+{a[34]}{a[35]}x*y}\\\\&=
\\simplify{({a[31]}{a[32]}+{a[34]}{a[42]})x+{a[31]}{a[33]}y+{a[34]}{a[35]}x*y}\\text{.}
\\end{align}
\\]

j)

\\[
\\begin{align}
\\simplify[terms]{{a[36]}a^2({a[37]}+{a[38]}b)+{a[39]}b^2({a[40]}a+{a[41]}b)}&=
\\simplify[!collectNumbers]{{a[37]}{a[36]}a^2+{a[38]}{a[36]}a^2b+{a[40]}{a[39]}a*b^2+{a[39]}{a[41]}b^3}\\\\&=
\\simplify{{a[37]}{a[36]}a^2+{a[38]}{a[36]}a^2b+{a[40]}{a[39]}a*b^2+{a[39]}{a[41]}b^3}\\text{.}
\\end{align}
\\]

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$\\simplify{{a[1]}({a[2]}x+{a[3]})}=$ [[0]]

It doesn't look like you've expanded - make sure you don't use any brackets in your answer.

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$\\simplify{{a[4]}({a[5]}x+{a[6]})}=$ [[0]]

It doesn't look like you've expanded - make sure you don't use any brackets in your answer.

$\\simplify{{a[7]}({a[8]}x^2+{a[9]}y)}=$ [[0]]

It doesn't look like you've expanded - make sure you don't use any brackets in your answer.

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$\\simplify{{a[10]}({a[11]}x^2+{a[12]}y)}=$ [[0]]

It doesn't look like you've expanded - make sure you don't use any brackets in your answer.

$\\simplify{{a[13]}x({a[14]}x^2+{a[15]}x+{a[16]})}=$ [[0]]

It doesn't look like you've expanded - make sure you don't use any brackets in your answer.

$\\simplify{{a[17]}x({a[18]}x^2+{a[19]}x+{a[20]})}=$ [[0]]

It doesn't look like you've expanded - make sure you don't use any brackets in your answer.

$\\simplify{{a[21]}x({a[22]}x^2+{a[23]}x)+{a[24]}x^2+{a[25]}x^3}=$ [[0]]

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$\\simplify{({a[26]}x^2+{a[27]}x^3)+{a[28]}x({a[29]}x^2+{a[30]}x)}=$ [[0]]

", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "x^2*{a[26]+a[28]*a[30]} + x^3*{a[28]*a[29]+a[27]}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "mustmatchpattern": {"pattern": "x^2*`+-$n`? + x^3*`+-$n`?", "partialCredit": 0, "message": "It doesn't look like you've expanded - make sure you don't use any brackets in your answer.", "nameToCompare": ""}, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "

$\\simplify{{a[31]}({a[32]}x+{a[33]}y)+{a[34]}x({a[42]}+{a[35]}y)}=$ [[0]]

", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "{a[31]*a[32]+a[34]*a[42]}x+{a[31]*a[33]}y+{a[34]*a[35]}x*y", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "mustmatchpattern": {"pattern": "x*(`+-$n) + y*(`+-$n) + x*y*(`+-$n)", "partialCredit": 0, "message": "", "nameToCompare": ""}, "valuegenerators": [{"name": "x", "value": ""}, {"name": "y", "value": ""}]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "

$\\simplify{{a[36]}a^2({a[37]}+{a[38]}b)+{a[39]}b^2({a[40]}a+{a[41]}b)}=$ [[0]]

It doesn't look like you've expanded - make sure you don't use any brackets in your answer.

", "nameToCompare": ""}, "valuegenerators": [{"name": "a", "value": ""}, {"name": "b", "value": ""}]}], "sortAnswers": false}], "type": "question"}, {"name": "Expand brackets and collect like terms", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Bradley Bush", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1521/"}, {"name": "Aiden McCall", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1592/"}], "variable_groups": [{"variables": ["a1", "b1", "c1"], "name": "B group"}, {"variables": ["a", "b", "c", "d", "f", "g", "h", "j"], "name": "Part a"}], "variables": {"c": {"templateType": "anything", "description": "", "definition": "repeat(random(2..10),5)", "name": "c", "group": "Part a"}, "c1": {"templateType": "anything", "description": "", "definition": "random(2..5)*2", "name": "c1", "group": "B group"}, "b1": {"templateType": "anything", "description": "", "definition": "random(2..10 except a1)", "name": "b1", "group": "B group"}, "d": {"templateType": "anything", "description": "", "definition": "repeat(random(2..33),6)", "name": "d", "group": "Part a"}, "f": {"templateType": "anything", "description": "", "definition": "repeat(random(2..20),7)", "name": "f", "group": "Part a"}, "j": {"templateType": "anything", "description": "", "definition": "repeat(random(2..20),9)", "name": "j", "group": "Part a"}, "h": {"templateType": "anything", "description": "", "definition": "repeat(random(2..20),7)", "name": "h", "group": "Part a"}, "a1": {"templateType": "anything", "description": "", "definition": "random(5..10)", "name": "a1", "group": "B group"}, "a": {"templateType": "anything", "description": "

random variables for part 1

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You must collect like terms to fully simplify.

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$\\var{c[0]}x+\\var{c[1]}x+\\var{c[2]}x=$ [[0]]

", "unitTests": [], "customName": ""}, {"extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "useCustomName": false, "showFeedbackIcon": true, "gaps": [{"showCorrectAnswer": true, "checkVariableNames": false, "useCustomName": false, "mustmatchpattern": {"nameToCompare": "", "partialCredit": 0, "pattern": "$n*x^2 + $n*x + $n", "message": "You haven't simplified: you still have two or more like terms that should be collected together."}, "unitTests": [], "answerSimplification": "all", "showFeedbackIcon": true, "type": "jme", "variableReplacementStrategy": "originalfirst", "failureRate": 1, "vsetRange": [0, 1], "variableReplacements": [], "customMarkingAlgorithm": "", "vsetRangePoints": 5, "valuegenerators": [{"value": "", "name": "x"}], "customName": "", "extendBaseMarkingAlgorithm": true, "checkingAccuracy": 0.001, "answer": "({a[1]}+{a[2]})x^2+({a[3]}+{a[4]})x+{a[0]}", "checkingType": "absdiff", "scripts": {}, "showPreview": true, "marks": 1}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "sortAnswers": false, "scripts": {}, "customMarkingAlgorithm": "", "marks": 0, "variableReplacements": [], "prompt": "

$\\var{a[1]}x^2+\\var{a[2]}x^2+\\var{a[3]}x+\\var{a[4]}x +\\var{a[0]}=$ [[0]]

You must condense your answer to fully simplify.

", "length": "0"}, "customMarkingAlgorithm": "", "vsetRangePoints": 5, "valuegenerators": [{"value": "", "name": "y"}], "customName": "", "extendBaseMarkingAlgorithm": true, "checkingAccuracy": 0.001, "answer": "({b[1]}+{b[2]}+{b[3]}+{b[4]}+{b[0]})y^5", "checkingType": "absdiff", "scripts": {}, "showPreview": true, "marks": 1}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "sortAnswers": false, "scripts": {}, "customMarkingAlgorithm": "", "marks": 0, "variableReplacements": [], "prompt": "

$\\var{b[0]}y^5+\\var{b[1]}y^5+\\var{b[2]}y^5+\\var{b[4]}y^5+\\var{b[3]}y^5=$ [[0]]

You must condense your answer to fully simplify.

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$\\var{d[0]}ab+\\var{d[1]}abc+\\var{d[2]}a+\\var{d[3]}b+\\var{d[4]}c+\\var{d[5]}abc=$ [[0]]

You must condense your answer to fully simplify.

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$\\var{f[0]}a^2b+\\var{f[1]}ab^2+\\var{f[2]}ab+\\var{f[3]}a^2b+\\var{f[4]}ab^2=$ [[0]]

You must condense your answer to fully simplify. *'s are not needed to indicate multiplication here.

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$\\var{g[0]}(\\var{g[1]}x+\\var{g[2]}y)+\\var{g[4]}x+\\var{g[5]}y=$ [[0]]

9You should not have brackets in your answer.

", "strings": ["(", ")"], "showStrings": true}, "variableReplacements": [], "vsetRange": [0, 1], "maxlength": {"partialCredit": 0, "message": "

You must condense your answer to fully simplify.

", "length": "0"}, "customMarkingAlgorithm": "", "vsetRangePoints": 5, "valuegenerators": [{"value": "", "name": "x"}, {"value": "", "name": "z"}], "customName": "", "extendBaseMarkingAlgorithm": true, "checkingAccuracy": 0.001, "answer": "({h[0]}{h[1]}+{h[4]})x^2+({h[0]}{h[2]})z*x+{h[3]}x+{h[5]}z^2+{h[6]}z", "checkingType": "absdiff", "scripts": {}, "showPreview": true, "marks": 1}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "sortAnswers": false, "scripts": {}, "customMarkingAlgorithm": "", "marks": 0, "variableReplacements": [], "prompt": "

$\\var{h[0]}x(\\var{h[1]}x+\\var{h[2]}z)+\\var{h[3]}x+\\var{h[6]}z+\\var{h[4]}x^2+\\var{h[5]}z^2=$ [[0]]

You must condense your answer to fully simplify.

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$\\var{j[0]}(\\var{j[1]}x-\\var{j[2]}y)+\\var{j[3]}(\\var{j[4]}x-\\var{j[5]}y)+\\var{j[6]}(\\var{j[7]}x-\\var{j[8]}y)=$ [[0]]

", "unitTests": [], "customName": ""}], "advice": "

When simplifying expressions, only terms of the same type or like terms can be added together.

Algebraic symbols or letters can be added together provided that they are raised to the same power. For example, we can add $x^2+x^2=2x^2$, but we cannot collect both $x^2$ and $x$ into one term.

a)

\\[
\\begin{align}
\\var{c[0]}x+\\var{c[1]}x+\\var{c[2]}x&=(\\var{c[0]}+\\var{c[1]}+\\var{c[2]})x\\\\
&=\\simplify{({c[0]}+{c[1]}+{c[2]})}x
\\end{align}
\\]

b)

\\[
\\begin{align}
\\var{a[1]}x^2+\\var{a[2]}x^2+\\var{a[3]}x+\\var{a[4]}x +\\var{a[0]}&=(\\var{a[1]}+\\var{a[2]})x^2+(\\var{a[3]}+\\var{a[4]})x +\\var{a[0]}\\\\
&=\\simplify{({a[1]}+{a[2]})}x^2+\\simplify{({a[3]}+{a[4]})}x+\\var{a[0]}
\\end{align}
\\]

c)

\\[
\\begin{align}
\\var{b[0]}y^5+\\var{b[1]}y^5+\\var{b[2]}y^5+\\var{b[4]}y^5+\\var{b[3]}y^5&=(\\var{b[0]}+\\var{b[1]}+\\var{b[2]}+\\var{b[4]}+\\var{b[3]})y^5\\\\
&=\\simplify{({b[1]}+{b[2]}+{b[3]}+{b[4]}+{b[0]})}y^5
\\end{align}
\\]

d)

\\[
\\begin{align}
\\var{d[0]}ab+\\var{d[1]}abc+\\var{d[2]}a+\\var{d[3]}b+\\var{d[4]}c+\\var{d[5]}abc
&=(\\var{d[1]}+\\var{d[5]})abc+\\var{d[0]}ab+\\var{d[2]}a+\\var{d[3]}b+\\var{d[4]}c\\\\
&=\\simplify{{d[1]}+{d[5]}}abc+\\var{d[0]}ab+\\var{d[2]}a+\\var{d[3]}b+\\var{d[4]}c
\\end{align}
\\]

e)

\\[
\\begin{align}
\\var{f[0]}a^2b+\\var{f[1]}ab^2+\\var{f[2]}ab+\\var{f[3]}a^2b+\\var{f[4]}ab^2
&=(\\var{f[0]}+\\var{f[3]})a^2b+(\\var{f[1]}+\\var{f[4]})ab^2+\\var{f[2]}ab\\\\
&=\\simplify{{f[0]}+{f[3]}}a^2b+\\simplify{{f[1]}+{f[4]}}ab^2+\\var{f[2]}ab
\\end{align}
\\]

f)

\\[
\\begin{align}
\\var{g[0]}(\\var{g[1]}x+\\var{g[2]}y)+\\var{g[4]}x+\\var{g[5]}y
&=(\\var{g[0]}\\times \\var{g[1]}+\\var{g[4]})x+(\\var{g[0]} \\times\\var{g[2]}+\\var{g[5]})y\\\\
&=(\\simplify{{g[0]}*{g[1]}}+\\var{g[4]})x+(\\simplify{{g[0]}*{g[2]}}+\\var{g[5]})y\\\\
&=\\simplify{{g[0]}*{g[1]}+{g[4]}}x+\\simplify{{g[0]}*{g[2]}+{g[5]}}y
\\end{align}
\\]

g)

\\[
\\begin{align}
\\var{h[0]}x(\\var{h[1]}x+\\var{h[2]}z)+\\var{h[3]}x+\\var{h[6]}z+\\var{h[4]}x^2+\\var{h[5]}z^2
&=(\\simplify[]{{h[0]}{h[1]}}+\\var{h[4]})x^2+(\\simplify[]{{h[0]}{h[2]}})zx+\\var{h[3]}x+\\var{h[5]}z^2+\\var{h[6]}z\\\\
&=(\\simplify{{h[0]}{h[1]}}+\\var{h[4]})x^2+(\\simplify[]{{h[0]}{h[2]}})zx+\\var{h[3]}x+\\var{h[5]}z^2+\\var{h[6]}z\\\\
&=\\simplify{{h[0]}*{h[1]}+{h[4]}}x^2+\\simplify{{h[0]}*{h[2]}}zx+\\simplify{{h[3]}x+{h[5]}}z^2+\\var{h[6]}z
\\end{align}
\\]

h)

\\[
\\begin{align}
\\var{j[0]}(\\var{j[1]}x-\\var{j[2]}y)+\\var{j[3]}(\\var{j[4]}x-\\var{j[5]}y)+\\var{j[6]}(\\var{j[7]}x-\\var{j[8]}y)
&= (\\simplify[]{{j[0]}{j[1]}}+\\simplify[]{{j[3]}{j[4]}}+\\simplify[]{{j[6]}{j[7]}})x-(\\simplify[]{{j[0]}{j[2]}}+\\simplify[]{{j[3]}{j[5]}}+\\simplify[]{{j[6]}{j[8]}})y\\\\
&= (\\simplify{{j[0]}{j[1]}}+\\simplify{{j[3]}{j[4]}}+\\simplify{{j[6]}{j[7]}})x-(\\simplify{{j[0]}{j[2]}}+\\simplify{{j[3]}{j[5]}}+\\simplify{{j[6]}{j[8]}})y\\\\
&= \\simplify{({j[0]}*{j[1]}+{j[4]*j[3]}+{j[6]}*{j[7]})x}-\\simplify{({j[0]}*{j[2]}+{j[5]}{j[3]}+{j[6]}*{j[8]})y}
\\end{align}
\\]

", "tags": ["collecting terms", "expanding brackets", "simplifying algebraic expressions", "simplifying expressions", "taxonomy"], "preamble": {"js": "", "css": ""}, "rulesets": {}, "functions": {}, "ungrouped_variables": [], "statement": "

For each expression below, collect like terms and expand brackets.

The * symbol is required between algebraic symbols, e.g. $5ab^2$ should be written 5*a*b^2.

", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

Eight expressions, of increasing complexity. The student must simplify them by expanding brackets and collecting like terms.

"}, "variablesTest": {"condition": "", "maxRuns": 100}}, {"name": "Solving linear inequalities", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Bradley Bush", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1521/"}], "metadata": {"description": "

In the first three parts, rearrange linear inequalities to make $x$ the subject.

In the last four parts, correctly give the direction of the inequality sign after rearranging an inequality.

", "licence": "Creative Commons Attribution 4.0 International"}, "ungrouped_variables": ["a", "b", "c"], "type": "question", "advice": "

As with regular linear equations, we aim to isolate the variable by subtracting any constants when dividing by the $x$ coefficient. The only major difference is that when we divide or multiply by a negative number, the inequality sign is reversed.

For example, the following inequality is true:

\\[ -3 \\lt -2 \\]

When we multiply both sides by $-2$, the inequality sign must reverse:

\\[ 6 \\gt 4 \\]

a)

To put $x$ on its own, we need to add $\\var{a[0]}$ to both sides of the inequality.

\\begin{align}
\\simplify{x-{a[0]}}&<\\var{a[1]}\\\\[1em]
\\var{x}&<\\simplify[]{{a[1]}+{a[0]}}\\\\[1em]
x&<\\simplify{({a[1]}+{a[0]})}\\text{.}
\\end{align}

b)

In this example we find $x$ by dividing both sides by the coefficient of $x$, $\\var{a[2]}$.

\\begin{align}
\\simplify{{a[2]}}x&<\\var{a[3]}\\\\[1em]
x&<\\simplify{{a[3]}/{a[2]}}\\text{.}
\\end{align}

c)

\\begin{align}
\\simplify{{a[6]}x-{a[4]}}&<\\var{a[5]}\\\\[1em]
\\var{a[6]}x&<\\var{a[5]}+\\var{a[4]} & \\text{Add } 8 \\text{ to get } x \\text{ on its own.}\\\\[1em]
x&<\\simplify[]{({a[5]}+{a[4]})/{a[6]}} & \\text{ Divide by } \\var{a[6]} \\text{.} \\\\[1em]
x&<\\simplify{({a[5]}+{a[4]})/{a[6]}}\\text{.}
\\end{align}

d)

In this example, take the constants to one side, and keep the $x$ term on the other. Divide through by the negative $x$-coefficient to find an inequality for $x$. Notice that where you divide (or multiply) an equality by a negative value, the inequality sign is reversed.

\\begin{align}
\\simplify{{-a[6]}x - {a[4]}} &< \\var{a[5]} \\\\[1em]
\\var{-a[6]}x &< \\var{a[5]} + \\var{a[4]} & \\text{Add } \\var{a[4]} \\text{ to both sides.} \\\\[1em]
x &> \\simplify[]{({a[5]}+{a[4]})/-{a[6]}} \\text{ Divide by } \\var{-a[6]} \\text{. The inequality is reversed.} \\\\[1em]
x &> \\simplify{({a[5]}+{a[4]})/-{a[6]}}\\text{.}\\\\
\\end{align}

e)

In this example, separate the constants and the $x$-term, then divide by the $x$-coefficient to find an inequality for $x$.

\\begin{align}
\\simplify{{b[0]}x-{b[1]}}&<\\simplify{{b[3]}-{b[2]}x}\\\\[1em]
\\simplify{({b[0]}+{b[2]})x}&<\\simplify{{b[3]}+{b[1]}}\\\\[1em]
x&<\\simplify{({b[3]}+{b[1]})/({b[0]}+{b[2]})}\\text{.}\\\\[1em]
\\end{align}

f)

In this example, separate the $x$-term from all other terms and remember to reverse the inequality when dividing by $\\simplify{{a[7]}-{b[4]}}$.

\\begin{align}
\\simplify{-{b[4]}x+{a[8]}a}&>\\simplify{{b[5]}+b-{a[7]}x}\\\\[1em]
\\simplify{{a[7]}-{b[4]}}x&>\\simplify{{b[5]}+b-{a[8]}a}\\\\[1em]
x&<\\simplify{(-{b[5]}-b+{a[8]}a)/({b[4]}-{a[7]})}\\text{.}\\\\[1em]
\\end{align}

In this example, a simple way to solve for $x$ is to divide by $-\\var{c}$ before rearranging the rest of the equation by subtracting $g$ from both sides.

\\begin{align}
\\simplify{-{c}(x+g)}&>\\simplify{6h-{c}{a[0]}}\\\\[1em]
\\simplify{(x+g)}&<\\simplify[]{6h/-{c}+{a[0]}}\\\\[1em]
x&<\\simplify[]{6h/-{c}+{a[0]}-g}\\\\[1em]
x&<\\simplify{{a[0]}-6h/{c}-g}\\text{.}
\\end{align}

", "variable_groups": [], "rulesets": {}, "statement": "

Solve the following linear inequalities by finding the set of possible values for $x$. State your answers as fractions where applicable.

", "parts": [{"scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "variableReplacements": [], "showCorrectAnswer": true, "marks": 0, "gaps": [{"variableReplacements": [], "scripts": {}, "vsetrangepoints": 5, "showCorrectAnswer": true, "checkingaccuracy": 0.001, "showFeedbackIcon": true, "checkvariablenames": false, "type": "jme", "answer": "({a[1]}+{a[0]})", "showpreview": true, "variableReplacementStrategy": "originalfirst", "checkingtype": "absdiff", "vsetrange": [0, 1], "marks": 1, "expectedvariablenames": []}], "showFeedbackIcon": true, "prompt": "

$\\simplify{x-{a[0]}<{a[1]}}$

$x<$ [[0]]

"}, {"scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "variableReplacements": [], "showCorrectAnswer": true, "marks": 0, "gaps": [{"variableReplacements": [], "scripts": {}, "vsetrangepoints": 5, "showCorrectAnswer": true, "checkingaccuracy": 0.001, "showFeedbackIcon": true, "checkvariablenames": false, "type": "jme", "answer": "{a[3]}/{a[2]}", "showpreview": true, "variableReplacementStrategy": "originalfirst", "checkingtype": "absdiff", "vsetrange": [0, 1], "marks": 1, "expectedvariablenames": [], "answersimplification": "all"}], "showFeedbackIcon": true, "prompt": "

$\\simplify{{a[2]}x<{a[3]}}$

$x<$ [[0]]

"}, {"scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "variableReplacements": [], "showCorrectAnswer": true, "marks": 0, "gaps": [{"variableReplacements": [], "scripts": {}, "vsetrangepoints": 5, "showCorrectAnswer": true, "checkingaccuracy": 0.001, "showFeedbackIcon": true, "checkvariablenames": false, "type": "jme", "answer": "({a[5]}+{a[4]})/{a[6]}", "showpreview": true, "variableReplacementStrategy": "originalfirst", "checkingtype": "absdiff", "vsetrange": [0, 1], "marks": 1, "expectedvariablenames": [], "answersimplification": "all"}], "showFeedbackIcon": true, "prompt": "

$\\simplify{{a[6]}x-{a[4]}<{a[5]}}$

$x<$ [[0]]

"}, {"scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "variableReplacements": [], "showCorrectAnswer": true, "marks": 0, "gaps": [{"variableReplacements": [], "scripts": {}, "vsetrangepoints": 5, "showCorrectAnswer": true, "checkingaccuracy": 0.001, "showFeedbackIcon": true, "checkvariablenames": false, "type": "jme", "answer": "({a[5]}+{a[4]})/-{a[6]}", "showpreview": true, "variableReplacementStrategy": "originalfirst", "checkingtype": "absdiff", "vsetrange": [0, 1], "marks": 1, "expectedvariablenames": [], "answersimplification": "all"}, {"shuffleChoices": false, "minMarks": 0, "showCorrectAnswer": true, "displayColumns": 0, "showFeedbackIcon": true, "choices": ["

", "

"], "scripts": {}, "distractors": ["", ""], "type": "1_n_2", "variableReplacementStrategy": "originalfirst", "maxMarks": 0, "marks": 0, "variableReplacements": [], "matrix": ["1", 0], "displayType": "dropdownlist"}], "showFeedbackIcon": true, "prompt": "

$\\simplify{{-a[6]}x-{a[4]}<{a[5]}}$

$x$ [[1]] [[0]]

"}, {"scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "variableReplacements": [], "showCorrectAnswer": true, "marks": 0, "gaps": [{"variableReplacements": [], "scripts": {}, "vsetrangepoints": 5, "showCorrectAnswer": true, "checkingaccuracy": 0.001, "showFeedbackIcon": true, "checkvariablenames": false, "type": "jme", "answer": "({b[3]}+{b[1]})/({b[0]}+{b[2]})", "showpreview": true, "variableReplacementStrategy": "originalfirst", "checkingtype": "absdiff", "vsetrange": [0, 1], "marks": 1, "expectedvariablenames": [], "answersimplification": "all"}, {"shuffleChoices": false, "minMarks": 0, "showCorrectAnswer": true, "displayColumns": 0, "showFeedbackIcon": true, "choices": ["

", "

"], "scripts": {}, "distractors": ["", ""], "type": "1_n_2", "variableReplacementStrategy": "originalfirst", "maxMarks": 0, "marks": 0, "variableReplacements": [], "matrix": [0, "1"], "displayType": "dropdownlist"}], "showFeedbackIcon": true, "prompt": "

$\\simplify{{b[0]}x-{b[1]}<{b[3]}-{b[2]}x}$

$x$ [[1]] [[0]]

"}, {"scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "variableReplacements": [], "showCorrectAnswer": true, "marks": 0, "gaps": [{"variableReplacements": [], "scripts": {}, "vsetrangepoints": 5, "showCorrectAnswer": true, "checkingaccuracy": 0.001, "showFeedbackIcon": true, "checkvariablenames": false, "type": "jme", "answer": "(-{b[5]}-b+{a[8]}a)/({b[4]}-{a[7]})", "showpreview": true, "variableReplacementStrategy": "originalfirst", "checkingtype": "absdiff", "vsetrange": [0, 1], "marks": 1, "expectedvariablenames": [], "answersimplification": "all"}, {"shuffleChoices": false, "minMarks": 0, "showCorrectAnswer": true, "displayColumns": 0, "showFeedbackIcon": true, "choices": ["

", "

"], "scripts": {}, "distractors": ["", ""], "type": "1_n_2", "variableReplacementStrategy": "originalfirst", "maxMarks": 0, "marks": 0, "variableReplacements": [], "matrix": [0, "1"], "displayType": "dropdownlist"}], "showFeedbackIcon": true, "prompt": "

$\\simplify{-{b[4]}x+{a[8]}a>{b[5]}+b-{a[7]}x}$

$x$ [[1]] [[0]]

"}, {"scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "variableReplacements": [], "showCorrectAnswer": true, "marks": 0, "gaps": [{"variableReplacements": [], "scripts": {}, "vsetrangepoints": 5, "showCorrectAnswer": true, "checkingaccuracy": 0.001, "showFeedbackIcon": true, "checkvariablenames": false, "type": "jme", "answer": "{a[0]}-6h/{c}-g", "showpreview": true, "variableReplacementStrategy": "originalfirst", "checkingtype": "absdiff", "vsetrange": [0, 1], "marks": 1, "expectedvariablenames": [], "answersimplification": "all"}, {"shuffleChoices": false, "minMarks": 0, "showCorrectAnswer": true, "displayColumns": 0, "showFeedbackIcon": true, "choices": ["

", "

"], "scripts": {}, "distractors": ["", ""], "type": "1_n_2", "variableReplacementStrategy": "originalfirst", "maxMarks": 0, "marks": 0, "variableReplacements": [], "matrix": [0, "1"], "displayType": "dropdownlist"}], "showFeedbackIcon": true, "prompt": "

$\\simplify{-{c}(x+g)>6h-{c}{a[0]}}$

$x$ [[1]] [[0]]

"}], "tags": ["inequalities", "linear inequalities", "solving linear inequalities", "taxonomy"], "preamble": {"css": "", "js": ""}, "functions": {}, "variables": {"b": {"description": "", "group": "Ungrouped variables", "definition": "repeat(random(11..20),10)", "name": "b", "templateType": "anything"}, "c": {"description": "", "group": "Ungrouped variables", "definition": "random(2,3,6)", "name": "c", "templateType": "anything"}, "a": {"description": "", "group": "Ungrouped variables", "definition": "repeat(random(2..10),10)", "name": "a", "templateType": "anything"}}, "variablesTest": {"maxRuns": 100, "condition": ""}}, {"name": "Extract common factors of polynomials", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Chris Graham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/369/"}, {"name": "Bradley Bush", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1521/"}], "advice": "

In order to factorise the expressions, the factors that make up each term in the expression need to be identified and, where these factors are the same for all terms in the expression, those factors can be taken outside the brackets. Stop when the remaining terms have no more common factors.

a)

Both terms have a common factor of $2$.

\\begin{align}
\\simplify{2{a[0]}x+2{b[0]}}&=
(\\simplify[]{2{a[0]}})x+2\\times\\var{b[0]}\\\\
&=\\simplify[]{2({a[0]}x+{b[0]})}
\\end{align}

b)

Both terms have common factors of $6$ and $y$.

\\begin{align}
\\simplify{6{a[1]}y+6{b[1]}y^2}&= 6 \\times \\var{a[1]} y + 6 \\times \\var{b[1]} y^2 \\\\
&= 6 \\times (\\simplify{{a[1]}y + {b[1]}y^2}) \\\\
&=6y(\\simplify[]{{a[1]}+{b[1]}y})
\\end{align}

c)

Both terms have common factors of $x$, $y$ and $z$.

\\begin{align}
\\simplify{{a[2]}x*y*z+{b[2]}x^2y^2z^2}&=\\var{a[2]} \\times xyz + \\var{b[2]} \\times xyz \\times xyz\\\\
&=xyz(\\var{a[2]} + \\var{b[2]} xyz)
\\end{align}

d)

All three terms have a common factor of $5$.

\\begin{align}
\\simplify{5{a[3]}d+5{b[3]}r+5m}&= 5 \\times \\var{a[3]} d+5 \\times \\var{b[3]} r + 5 m \\\\
&=\\simplify[]{5({a[3]}d+{b[3]}r+m)}
\\end{align}

e)

All the terms have common factors of $6$, $c$ and $d$.

\\begin{align}
\\simplify{6{a[4]}cd^2+6{b[4]}c^2d+6{c[1]}c^2d^2} &= 6 \\times \\var{a[4]} c d^2 \\;+\\; 6 \\times \\var{b[4]} c^2 d \\;+\\; 6 \\times \\var{c[1]} c^2 d^2 \\\\
&= 6(\\var{a[4]} c d^2 + \\var{b[4]} c^2 d + \\var{c[1]} c^2 d^2) \\\\
&=6cd(\\var{a[4]}d+\\var{b[4]}c+\\var{c[1]}cd)
\\end{align}

", "statement": "

An expression can be factorised by finding common factors of each term in the expression.

Completely factorise the following expressions by finding their common factors.

Make sure that you include a multiplication symbol * between each algebraic variable, and before brackets, e.g. a*b*(x+1) instead of ab(x+1). Otherwise, the system might not accept your answer.

", "preamble": {"js": "question.mark_factorised = function(part) {\n var match = Numbas.jme.display.matchExpression;\n var unwrap = Numbas.jme.unwrapValue;\n var getCommutingTerms = Numbas.jme.display.getCommutingTerms;\n var matchTree = Numbas.jme.display.matchTree;\n\n var expr = part.studentAnswer;\n\n // is the student's answer in the form `factors*(terms)`?\n var m = match('m_all(m_any(m_number,m_type(name),m_type(name)^m_number))*(m_all(??)+m_nothing);rest', expr, true);\n if(!m) {\n part.multCredit(0,\"You don't seem to have extracted a common factor.\");\n return;\n }\n \n var terms = getCommutingTerms(m.rest,'+').terms;\n \n var gcd; // greatest common divisor of the coefficients of each term\n var min_degrees = {}; // minimum degree of each variable across the terms - if greater than 0, that variable is a common factor\n \n // for each term, collect the scalar part and the degree of each of the variables\n terms.map(function(t,i) { \n var factors = getCommutingTerms(t,'*').terms;\n var degrees = {};\n var n = 1;\n factors.forEach(function(f) {\n var tok = f.tok;\n var scalar = 1;\n var name = null;\n var degree = 0;\n var m = Numbas.jme.display.matchTree(Numbas.jme.compile('m_any(m_number;n, m_type(name);name, m_number;n*m_type(name);name, m_type(name);name^(m_number;degree), m_number;n*m_type(name);name^m_number;degree )'),f,true)\n\n console.log(Numbas.jme.display.treeToJME(f));\n\n if(!m) {\n console.log(' no match');\n return;\n }\n console.log(' match',m.n,m.name,m.degree);\n if(m.n) {\n n *= unwrap(m.n.tok);\n }\n if(m.name) {\n name = m.name.tok.name;\n if(m.degree) {\n degree = unwrap(m.degree.tok);\n } else {\n degree = 1;\n }\n degrees[name] = (degrees[name] || 0) + degree;\n }\n });\n console.log(Numbas.jme.display.treeToJME(t), degrees, n);\n \n if(i==0) {\n gcd = n;\n } else {\n gcd = Numbas.math.gcd(gcd,n);\n }\n \n Object.keys(degrees).forEach(function(k) {\n if(i==0) {\n min_degrees[k] = degrees[k];\n } else {\n min_degrees[k] = Math.min(min_degrees[k] || 0, degrees[k]);\n }\n });\n Object.keys(min_degrees).forEach(function(k) {\n if(degrees[k]===undefined) {\n min_degrees[k] = 0;\n }\n });\n \n });\n \n var common_factors = [];\n if(gcd>1) {\n common_factors.push(gcd);\n }\n Object.keys(min_degrees).filter(function(k){return min_degrees[k]>0}).forEach(function(k) {\n common_factors.push(k+'^'+min_degrees[k])\n });\n \n if(common_factors.length==0) {\n return true;\n } else {\n part.multCredit(0,\"Your terms still have a common factor of $\"+Numbas.jme.display.exprToLaTeX(common_factors.join('*'),'all',Numbas.jme.builtinScope)+\"$.\");\n }\n};", "css": ""}, "variables": {"x7": {"templateType": "anything", "name": "x7", "description": "", "group": "Ungrouped variables", "definition": "random(1..5)"}, "a": {"templateType": "anything", "name": "a", "description": "

Vector of every other random prime number

", "group": "Ungrouped variables", "definition": "repeat(random([3, 11, 17, 29, 37, 43] except b),50)"}, "x2": {"templateType": "anything", "name": "x2", "description": "", "group": "Ungrouped variables", "definition": "random(1..5)"}, "x3": {"templateType": "anything", "name": "x3", "description": "", "group": "Ungrouped variables", "definition": "random(1..5 except [x2])"}, "b": {"templateType": "anything", "name": "b", "description": "

Vector of the other every other random prime number

", "group": "Ungrouped variables", "definition": "repeat(random(2, 7, 13, 23, 31, 41, 53),50)"}, "c": {"templateType": "anything", "name": "c", "description": "

extra primes for when you need a third constant

", "group": "Ungrouped variables", "definition": "repeat(random([ 5, 19, 47] ),50)"}, "x1": {"templateType": "anything", "name": "x1", "description": "", "group": "Ungrouped variables", "definition": "random(1..5)"}, "x4": {"templateType": "anything", "name": "x4", "description": "", "group": "Ungrouped variables", "definition": "random(-5..-1)\n\n"}, "x5": {"templateType": "anything", "name": "x5", "description": "", "group": "Ungrouped variables", "definition": "random(1..5)"}, "x6": {"templateType": "anything", "name": "x6", "description": "", "group": "Ungrouped variables", "definition": "random(1..5)"}}, "tags": ["common factors", "common factors of linear algebraic equations", "common factors of quadratic equations", "finding common factors", "Linear equations", "linear equations", "quadratic equations", "Quadratic Equations", "Quadratic equations", "taxonomy"], "ungrouped_variables": ["a", "b", "c", "x1", "x2", "x3", "x4", "x5", "x6", "x7"], "functions": {}, "metadata": {"description": "

Factorise polynomials by identifying common factors. The first expression has a constant common factor; the rest have common factors involving variables.

", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "variable_groups": [], "parts": [{"sortAnswers": false, "variableReplacements": [], "gaps": [{"variableReplacements": [], "answer": "2({a[0]}x+{b[0]})", "showPreview": true, "expectedVariableNames": [], "unitTests": [], "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "scripts": {"mark": {"order": "after", "script": "question.mark_factorised(this);"}}, "showCorrectAnswer": true, "failureRate": 1, "type": "jme", "checkVariableNames": false, "showFeedbackIcon": true, "customMarkingAlgorithm": "", "marks": 1, "vsetRange": [0, 1], "variableReplacementStrategy": "originalfirst", "vsetRangePoints": 5}], "unitTests": [], "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "scripts": {}, "showCorrectAnswer": true, "type": "gapfill", "prompt": "

$\\simplify{{2*a[0]}x+{2*b[0]}}=$ [[0]]

", "customMarkingAlgorithm": "", "marks": 0, "variableReplacementStrategy": "originalfirst"}, {"sortAnswers": false, "variableReplacements": [], "gaps": [{"variableReplacements": [], "answer": "6*y*({a[1]}+{b[1]}*y)", "showPreview": true, "expectedVariableNames": [], "unitTests": [], "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "scripts": {"mark": {"order": "after", "script": "question.mark_factorised(this);"}}, "showCorrectAnswer": true, "failureRate": 1, "type": "jme", "checkVariableNames": false, "showFeedbackIcon": true, "customMarkingAlgorithm": "", "marks": 1, "vsetRange": [0, 1], "variableReplacementStrategy": "originalfirst", "vsetRangePoints": 5}], "unitTests": [], "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "scripts": {}, "showCorrectAnswer": true, "type": "gapfill", "prompt": "

$\\simplify{6{a[1]}y+6{b[1]}y^2}=$ [[0]]

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$\\simplify{{a[2]}x*y*z+{b[2]}x^2y^2z^2}=$ [[0]]

", "customMarkingAlgorithm": "", "marks": 0, "variableReplacementStrategy": "originalfirst"}, {"sortAnswers": false, "variableReplacements": [], "gaps": [{"variableReplacements": [], "answer": "5({a[3]}d+{b[3]}r+m)", "showPreview": true, "expectedVariableNames": [], "unitTests": [], "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "scripts": {"mark": {"order": "after", "script": "question.mark_factorised(this);"}}, "showCorrectAnswer": true, "failureRate": 1, "type": "jme", "checkVariableNames": false, "showFeedbackIcon": true, "customMarkingAlgorithm": "", "marks": 1, "vsetRange": [0, 1], "variableReplacementStrategy": "originalfirst", "vsetRangePoints": 5}], "unitTests": [], "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "scripts": {}, "showCorrectAnswer": true, "type": "gapfill", "prompt": "

$\\simplify{5{a[3]}d+5{b[3]}r+5m}=$ [[0]]

", "customMarkingAlgorithm": "", "marks": 0, "variableReplacementStrategy": "originalfirst"}, {"sortAnswers": false, "variableReplacements": [], "gaps": [{"variableReplacements": [], "answer": "6c*d*({a[4]}d+{b[4]}c+{c[1]}c*d)", "showPreview": true, "expectedVariableNames": [], "unitTests": [], "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "scripts": {"mark": {"order": "after", "script": "question.mark_factorised(this);"}}, "showCorrectAnswer": true, "failureRate": 1, "type": "jme", "checkVariableNames": false, "showFeedbackIcon": true, "customMarkingAlgorithm": "", "marks": 1, "vsetRange": [0, 1], "variableReplacementStrategy": "originalfirst", "vsetRangePoints": 5}], "unitTests": [], "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "scripts": {}, "showCorrectAnswer": true, "type": "gapfill", "prompt": "

$\\simplify{6{a[4]}c*d^2+6{b[4]}c^2d+6{c[1]}c^2d^2}=$ [[0]]

", "customMarkingAlgorithm": "", "marks": 0, "variableReplacementStrategy": "originalfirst"}], "variablesTest": {"condition": "", "maxRuns": "1000"}, "rulesets": {}}, {"name": "Dividing a polynomial with remainders, using algebraic division", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Elliott Fletcher", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1591/"}], "type": "question", "statement": "", "variablesTest": {"condition": "", "maxRuns": 100}, "variables": {"const": {"group": "Ungrouped variables", "description": "

Constant term in part b.

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Dividing term.

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Coefficient of x in part b.

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Positive coefficient of x^3

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Coefficient of x.

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Constant term in the divisor in part b.

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Coefficient of x^2.

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Free coefficient.

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Coefficient of x in divisor in part b.

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Coefficient of x^2 in part b.

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Used to simplify calculation.

", "templateType": "anything", "name": "n", "definition": "((b)-(a)*(f))*f"}}, "functions": {}, "tags": ["algebraic division", "division with remainder", "Long division", "long division", "polynomials", "Polynomials", "taxonomy"], "variable_groups": [], "parts": [{"checkingtype": "absdiff", "scripts": {}, "showpreview": true, "variableReplacementStrategy": "originalfirst", "vsetrangepoints": 5, "showCorrectAnswer": true, "showFeedbackIcon": true, "prompt": "

Find the remainder when $g(x) = \\simplify{{a}x^3+{b}x^2+{c}x+{d}}$ is divided by $(\\simplify{x+{f}})$.

", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "expectedvariablenames": [], "marks": 1, "variableReplacements": [], "answer": "{{d}-{c}*{f}+{b}*{f}^2-{a}*{f}^3}", "checkvariablenames": false, "type": "jme"}, {"checkingtype": "absdiff", "scripts": {}, "showpreview": true, "variableReplacementStrategy": "originalfirst", "vsetrangepoints": 5, "showCorrectAnswer": true, "showFeedbackIcon": true, "prompt": "

Find the remainder when $p(x) = \\simplify{x^3+{coef_x2}x^2+{coef_x}x+{const}}$ is divided by $\\simplify{x^2+{coef2_x}x+{const2}}$.

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Find the remainder when dividing two polynomials, by algebraic long division.

"}, "preamble": {"css": "", "js": ""}, "advice": "

Algebraic long division is a method of presenting the division of polynomials.

For example, we can use long division to write

\\[\\displaystyle\\frac{\\simplify{x^3+4x^2+5x+8}}{\\simplify{x-2}} = \\simplify{x-2}\\;\\overline{)\\simplify{x^3+4x^2+5x+8}}.\\]

Notice that we write the denominator - the polynomial we're dividing by - to the left of the numerator - the polynomial to be divided - with a division line between them.

The process of long division goes as follows:

Divide the first term of the numerator by the first term of the denominator; this becomes the first term of the answer.
Multiply the denominator by the term you just obtained, and write the result directly below the numerator.
Subtract this expression from the numerator to obtain a new polynomial representing the remainder.

We then repeat these same steps with the remainder until we are only left with an integer or a polynomial of lower degree than the denominator.

a)

The long division illustration will be shown for each step in the following calculation:

To divide $\\simplify[all,!noleadingMinus]{{a}x^3+{b}x^2+{c}x+{d}}$ by $(\\simplify[all,!noleadingMinus]{x+{f}})$, using algebraic division, we must first divide the first term of the polynomial by the first term of the divisor, which in this case is $x$, so that

\\[
\\begin{align}
\\simplify{{a}x^3} \\div x &= \\simplify{{a}x^2}. & &\\,\\,\\simplify{{a}x^2}\\\\
&& \\simplify[all,!noleadingMinus]{x+{f}} \\; &\\overline{)\\simplify[all,!noleadingMinus]{{a}x^3+{b}x^2+{c}x+{d}}}\\\\
\\end{align}
\\]

Next, we multiply $(\\simplify{x+{f}})$ by $\\simplify{{a}x^2}$, giving us

\\[
\\begin{align}
\\simplify{{a}x^2} \\times (\\simplify{x+{f}}) &= \\simplify[all,!noleadingMinus]{{a}x^3+{a}{f}x^2}. &&\\,\\,\\simplify{{a}x^2}\\\\
&&\\simplify[all,!noleadingMinus]{x+{f}} \\; &\\overline{)\\simplify[all,!noleadingMinus]{{a}x^3+{b}x^2+{c}x+{d}}}\\\\
&&&\\;\\,
\\simplify[all,!noleadingMinus]{{a}x^3+{a}{f}x^2}\\\\
\\end{align}
\\]

Then, we subtract this expression from the first two terms of the polynomial, which gives

\\[
\\begin{align}
(\\simplify[all,!noleadingMinus]{{a}x^3+{b}x^2}) - (\\simplify{{a}x^3+{a}{f}x^2}) &= \\simplify[all,!noleadingMinus]{({b-a*f})x^2}. &&\\,\\,\\simplify{{a}x^2}\\\\
&&\\simplify[all,!noleadingMinus]{x+{f}} \\; &\\overline{)\\simplify[all,!noleadingMinus]{{a}x^3+{b}x^2+{c}x+{d}}}\\\\
&&&\\;\\,
\\simplify[all,!noleadingMinus]{{a}x^3+{a}{f}x^2}\\\\
&&&\\;\\qquad\\quad
\\overline{\\simplify[all,!noleadingMinus]{({b-a*f})x^2+{c}x+{d}}}\\text{.}\\\\
\\end{align}
\\]

We bring the $\\simplify[all,!noleadingMinus]{{c}x+{d}}$ down from the line above, as these terms have not yet been used in the calculation.

We then repeat the previous steps for this new polynomial.

We divide the first term of the new polynomial by the first term of the divisor.

Dividing $\\simplify[all,!noleadingMinus]{({b-a*f})x^2}$ by $x$, gives

\\[
\\begin{align}
\\simplify[all,!noleadingMinus]{({b-a*f})x^2} \\div x &= \\simplify[all,!noleadingMinus]{({b-a*f})x}. &&\\,\\,\\simplify{{a}x^2+({b}-{a}{f})x}\\\\
&&\\simplify[all,!noleadingMinus]{x+{f}} \\; &\\overline{)\\simplify[all,!noleadingMinus]{{a}x^3+{b}x^2+{c}x+{d}}}\\\\
&&&\\;\\,
\\simplify[all,!noleadingMinus]{{a}x^3+{a}{f}x^2}\\\\
&&&\\;\\;\\qquad\\quad
\\overline{\\simplify[all,!noleadingMinus]{({b-a*f})x^2+{c}x+{d}}}\\text{.}\\\\
\\end{align}
\\]

Multiplying $(\\simplify{x+{f}})$ by $\\simplify{({b-a*f})x}$ gives us

\\[
\\begin{align}
\\simplify{({b-a*f})x} \\times (\\simplify{x+{f}}) &= \\simplify[all,!noleadingMinus]{({b-a*f})x^2+{n}x}. &&\\,\\,\\simplify{{a}x^2+({b}-{a}{f})x}\\\\
&&\\simplify[all,!noleadingMinus]{x+{f}} \\; &\\overline{)\\simplify[all,!noleadingMinus]{{a}x^3+{b}x^2+{c}x+{d}}}\\\\
&&&\\;\\,
\\simplify[all,!noleadingMinus]{{a}x^3+{a}{f}x^2}\\\\
&&&\\;\\;\\qquad\\quad
\\overline{\\simplify[all,!noleadingMinus]{({b-a*f})x^2+{c}x+{d}}}\\\\
&&&\\;\\;\\qquad\\quad
\\simplify[all,!noleadingMinus]{({b-a*f})x^2+{n}x}\\\\
\\end{align}
\\]

Now, subtracting $\\simplify[all,!noleadingMinus]{({b-a*f})x^2+{n}x}$ from $\\simplify[all,!noleadingMinus]{({b-a*f})x^2+{c}x+{d}}$ so that

\\[
\\begin{align}
(\\simplify[all,!noleadingMinus]{({b-a*f})x^2+{c}x+{d}}) - (\\simplify[all,!noleadingMinus]{({b-a*f})x^2+{n}x}) &= \\simplify{({c-n})x+{d}}. &&\\,\\,\\simplify{{a}x^2+({b}-{a}{f})x}\\\\
&&\\simplify[all,!noleadingMinus]{x+{f}} \\; &\\overline{)\\simplify[all,!noleadingMinus]{{a}x^3+{b}x^2+{c}x+{d}}}\\\\
&&&\\;\\,
\\simplify[all,!noleadingMinus]{{a}x^3+{a}{f}x^2}\\\\
&&&\\;\\;\\qquad\\quad
\\overline{\\simplify[all,!noleadingMinus]{({b-a*f})x^2+{c}x+{d}}}\\\\
&&&\\;\\;\\qquad\\quad
\\simplify[all,!noleadingMinus]{({b-a*f})x^2+{n}x}\\\\
&&&\\;\\;\\;\\qquad\\quad\\quad\\quad\\quad
\\overline{\\simplify[all,!noleadingMinus]{{{c}-{b}*{f}+{a}*{f}^2}x+{d}}}\\text{.}\\\\
\\end{align}
\\]

Finally, we have to repeat this method one more time.

We divide the first term of the new polynomial by the first term of the divisor.

Hence, we divide $\\simplify{({c-n})x}$ by $x$, so that

\\[
\\begin{align}
\\simplify{({c-n})x} \\div x &= \\simplify{{c-n}}. &&\\,\\,\\simplify{{a}x^2+({b}-{a}{f})x+{{c}-{b}*{f}+{a}*{f}^2}}\\\\
&&\\simplify[all,!noleadingMinus]{x+{f}} \\; &\\overline{)\\simplify[all,!noleadingMinus]{{a}x^3+{b}x^2+{c}x+{d}}}\\\\
&&&\\;\\,
\\simplify[all,!noleadingMinus]{{a}x^3+{a}{f}x^2}\\\\
&&&\\;\\;\\qquad\\quad
\\overline{\\simplify[all,!noleadingMinus]{({b-a*f})x^2+{c}x+{d}}}\\\\
&&&\\;\\;\\qquad\\quad
\\simplify[all,!noleadingMinus]{({b-a*f})x^2+{n}x}\\\\
&&&\\;\\;\\;\\qquad\\quad\\quad\\quad\\quad
\\overline{\\simplify[all,!noleadingMinus]{{{c}-{b}*{f}+{a}*{f}^2}x+{d}}}\\\\
\\end{align}
\\]

Then, multiply $(\\simplify{x+{f}})$ by $\\simplify{{c-n}}$, giving us

\\[
\\begin{align}
\\simplify{{c-n}} \\times (\\simplify{x+{f}}) &= \\simplify{({c-n})x+{c-n}{f}}. &&\\,\\,\\simplify{{a}x^2+({b}-{a}{f})x+{{c}-{b}*{f}+{a}*{f}^2}}\\\\
&&\\simplify[all,!noleadingMinus]{x+{f}} \\; &\\overline{)\\simplify[all,!noleadingMinus]{{a}x^3+{b}x^2+{c}x+{d}}}\\\\
&&&\\;\\,
\\simplify[all,!noleadingMinus]{{a}x^3+{a}{f}x^2}\\\\
&&&\\;\\;\\qquad\\quad
\\overline{\\simplify[all,!noleadingMinus]{({b-a*f})x^2+{c}x+{d}}}\\\\
&&&\\;\\;\\qquad\\quad
\\simplify[all,!noleadingMinus]{({b-a*f})x^2+{n}x}\\\\
&&&\\;\\;\\qquad\\quad\\quad\\quad\\quad
\\overline{\\simplify[all,!noleadingMinus]{{{c}-{b}*{f}+{a}*{f}^2}x+{d}}}\\\\
&&&\\;\\;\\qquad\\quad\\quad\\quad\\quad
\\simplify[all,!noleadingMinus]{{{c}-{b}*{f}+{a}*{f}^2}x+{{c}*{f}-{b}*{f}^2+{a}*{f}^3}}\\\\
\\end{align}
\\]

Lastly, subtract $\\simplify[all,!noleadingMinus]{({c-n})x+{c-n}{f}}$ from $\\simplify[all,!noleadingMinus]{({c-n})x+{d}}$, so that

\\[
\\begin{align}
(\\simplify[all,!noleadingMinus]{({c-n})x+{d}}) - (\\simplify[all,!noleadingMinus]{({c-n})x+{c-n}{f}}) &= \\simplify[all,!noleadingMinus]{{{d}-({c-n})*{f}}}. &&\\,\\,\\simplify{{a}x^2+({b}-{a}{f})x+{{c}-{b}*{f}+{a}*{f}^2}}\\\\
&&\\simplify[all,!noleadingMinus]{x+{f}} \\; &\\overline{)\\simplify[all,!noleadingMinus]{{a}x^3+{b}x^2+{c}x+{d}}}\\\\
&&&\\;\\,
\\simplify[all,!noleadingMinus]{{a}x^3+{a}{f}x^2}\\\\
&&&\\;\\;\\qquad\\quad
\\overline{\\simplify[all,!noleadingMinus]{({b-a*f})x^2+{c}x+{d}}}\\\\
&&&\\;\\;\\qquad\\quad
\\simplify[all,!noleadingMinus]{({b-a*f})x^2+{n}x}\\\\
&&&\\;\\;\\qquad\\quad\\quad\\quad\\quad
\\overline{\\simplify[all,!noleadingMinus]{{{c}-{b}*{f}+{a}*{f}^2}x+{d}}}\\\\
&&&\\;\\;\\qquad\\quad\\quad\\quad\\quad
\\simplify[all,!noleadingMinus]{{{c}-{b}*{f}+{a}*{f}^2}x+{{c}*{f}-{b}*{f}^2+{a}*{f}^3}}\\\\
&&&\\;\\;\\;\\qquad\\qquad\\quad\\quad\\quad\\;
\\overline{\\simplify[all,!noleadingMinus]{{{d}-{c}*{f}+{b}*{f}^2-{a}*{f}^3}}.}
\\end{align}
\\]

b)

The long division illustration will be shown for each step in the following calculation:

To divide $\\simplify[all,!noleadingMinus]{x^3+{coef_x2}x^2+{coef_x}x+{const}}$ by $\\simplify[all,!noleadingMinus]{x^2+{coef2_x}x+{const2}}$, using algebraic division, we must first divide the first term of the polynomial by the first term of the divisor, which in this case is $x^2$, so that

\\[
\\begin{align}
\\simplify{x^3} \\div x^2 &= \\simplify{x}. & &\\,\\,\\simplify{x}\\\\
&& \\simplify[all,!noleadingMinus]{x^2+{coef2_x}x+{const2}} \\; &\\overline{)\\simplify[all,!noleadingMinus]{x^3+{coef_x2}x^2+{coef_x}x+{const}}}\\\\
\\end{align}
\\]

Next, we multiply $\\simplify{x^2+{coef2_x}x+{const2}}$ by $\\simplify{x}$, giving us

\\[
\\begin{align}
\\simplify{x} \\times (\\simplify{x^2+{coef2_x}x+{const2}}) &= \\simplify[all,!noleadingMinus]{x^3+{coef2_x}x^2+{const2}x}. &&\\,\\,\\simplify{x}\\\\
&&\\simplify[all,!noleadingMinus]{x^2+{coef2_x}x+{const2}} \\; &\\overline{)\\simplify[all,!noleadingMinus]{x^3+{coef_x2}x^2+{coef_x}x+{const}}}\\\\
&&&\\;\\,
\\simplify[all,!noleadingMinus]{x^3+{coef2_x}x^2+{const2}x}\\\\
\\end{align}
\\]

Then, we subtract this expression from the first three terms of the polynomial, which gives

\\[
\\begin{align}
(\\simplify[all,!noleadingMinus]{x^3+{coef_x2}x^2+{coef_x}x}) - (\\simplify{x^3+{coef2_x}x^2+{const2}x}) &= \\simplify[all,!noleadingMinus]{{coef_x2-coef2_x}x^2+{coef_x-const2}x}. &&\\,\\,\\simplify{x}\\\\
&&\\simplify[all,!noleadingMinus]{x^2+{coef2_x}x+{const2}}\\; &\\overline{)\\simplify[all,!noleadingMinus]{x^3+{coef_x2}x^2+{coef_x}x+{const}}}\\\\
&&&\\;\\,
\\simplify[all,!noleadingMinus]{x^3+{coef2_x}x^2+{const2}x}\\\\
&&&\\;\\qquad\\quad
\\overline{\\simplify[all,!noleadingMinus]{{coef_x2-coef2_x}x^2+{coef_x-const2}x+{const}}}\\\\
\\end{align}
\\]

Note that we bring the $\\simplify[all,!noleadingMinus]{{const}}$ down from the line above, as this term has not yet been used in the calculation.

We then repeat the previous steps for this new polynomial.

We divide the first term of the new polynomial by the first term of the divisor.

Dividing $\\simplify[all,!noleadingMinus]{{coef_x2-coef2_x}x^2}$ by $x^2$ gives,

\\[
\\begin{align}
\\simplify[all,!noleadingMinus]{{coef_x2-coef2_x}x^2} \\div x^2 &= \\simplify[all,!noleadingMinus]{{coef_x2-coef2_x}}. &&\\,\\,\\simplify{x+{coef_x2-coef2_x}}\\\\
&&\\simplify[all,!noleadingMinus]{x^2+{coef2_x}x+{const2}}\\; &\\overline{)\\simplify[all,!noleadingMinus]{x^3+{coef_x2}x^2+{coef_x}x+{const}}}\\\\
&&&\\;\\,
\\simplify[all,!noleadingMinus]{x^3+{coef2_x}x^2+{const2}x}\\\\
&&&\\;\\;\\qquad\\quad
\\overline{\\simplify[all,!noleadingMinus]{{coef_x2-coef2_x}x^2+{coef_x-const2}x+{const}}}\\\\
\\end{align}
\\]

Multiplying $\\simplify{x^2+{coef2_x}x+{const2}}$ by $\\simplify{{coef_x2-coef2_x}}$ gives us

\\[
\\begin{align}
(\\simplify{x^2+{coef2_x}x+{const2}}) \\times \\simplify{{coef_x2-coef2_x}} &= \\simplify[all,!noleadingMinus]{{coef_x2-coef2_x}x^2+{coef2_x*(coef_x2-coef2_x)}x+{const2*(coef_x2-coef2_x)}}. &&\\,\\,\\simplify{x+{coef_x2-coef2_x}}\\\\
&&\\simplify[all,!noleadingMinus]{x^2+{coef2_x}x+{const2}}\\; &\\overline{)\\simplify[all,!noleadingMinus]{x^3+{coef_x2}x^2+{coef_x}x+{const}}}\\\\
&&&\\;\\,
\\simplify[all,!noleadingMinus]{x^3+{coef2_x}x^2+{const2}x}\\\\
&&&\\;\\;\\qquad\\quad
\\overline{\\simplify[all,!noleadingMinus]{{coef_x2-coef2_x}x^2+{coef_x-const2}x+{const}}}\\\\
&&&\\;\\;\\qquad\\quad
\\simplify[all,!noleadingMinus]{{coef_x2-coef2_x}x^2+{coef2_x*(coef_x2-coef2_x)}x+{const2*(coef_x2-coef2_x)}}.
\\end{align}
\\]

Now, subtracting $\\simplify[all,!noleadingMinus]{{coef_x2-coef2_x}x^2+{coef2_x*(coef_x2-coef2_x)}x+{const2*(coef_x2-coef2_x)}}$ from $\\simplify[all,!noleadingMinus]{{coef_x2-coef2_x}x^2+{coef_x-const2}x+{const}}$ so that

\\[
\\begin{align}
(\\simplify[all,!noleadingMinus]{{coef_x2-coef2_x}x^2+{coef_x-const2}x+{const}})- (\\simplify[all,!noleadingMinus]{{coef_x2-coef2_x}x^2+{coef2_x*(coef_x2-coef2_x)}x+{const2*(coef_x2-coef2_x)}}) &= \\simplify{({coef_x-const2-coef2_x*(coef_x2-coef2_x)})x+{const-const2*(coef_x2-coef2_x)}}. &&\\,\\,\\simplify{x+{coef_x2-coef2_x}}\\\\
&&\\simplify[all,!noleadingMinus]{x^2+{coef2_x}x+{const2}}\\; &\\overline{)\\simplify[all,!noleadingMinus]{x^3+{coef_x2}x^2+{coef_x}x+{const}}}\\\\
&&&\\;\\,
\\simplify[all,!noleadingMinus]{x^3+{coef2_x}x^2+{const2}x}\\\\
&&&\\;\\;\\qquad\\quad
\\overline{\\simplify[all,!noleadingMinus]{{coef_x2-coef2_x}x^2+{coef_x-const2}x+{const}}}\\\\
&&&\\;\\;\\qquad\\quad
\\simplify[all,!noleadingMinus]{{coef_x2-coef2_x}x^2+{coef2_x*(coef_x2-coef2_x)}x+{const2*(coef_x2-coef2_x)}}\\\\
&&&\\;\\;\\;\\qquad\\quad\\quad\\quad\\quad
\\overline{\\simplify{({coef_x-const2-coef2_x*(coef_x2-coef2_x)})x+{const-const2*(coef_x2-coef2_x)}}}\\text{.}\\\\
\\end{align}
\\]

"}, {"name": "Use the factor theorem to identify factors of a polynomial", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Elliott Fletcher", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1591/"}], "rulesets": {}, "functions": {}, "ungrouped_variables": ["a", "b", "c", "d", "coef1_x3", "coef1_x2", "coef1_x", "const", "coef2_x3", "coef2_x2", "coef2_x", "coef3_x3", "coef3_x2", "coef3_x"], "metadata": {"description": "

Apply the factor theorem to check which of a list of linear polynomials are factors of another polynomial.

", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "variable_groups": [], "advice": "

To find the factors of the polynomial $f(x) = \\simplify{x^3+({a}+{b}+{c})x^2+({a}{b}+{a}{c}+{b}{c})x+{a}{b}{c}}$, we use the factor theorem.

If $f(x)$ is a polynomial and $f(p) = 0$, then $(x-p)$ is a factor of $f(x)$.

If $(\\simplify{(x+{a})})$ is a factor of $f(x)$ then by the factor theorem, $f(\\simplify{-{a}}) = 0$.

We see that

\\[
\\begin{align}
f(\\simplify{-{a}}) &= \\simplify[all,!collectNumbers]{{coef1_x3}+{coef1_x2}+{coef1_x}+{const}}\\\\
&= \\simplify{{coef1_x3}+{coef1_x2}+{coef1_x}+{const}}.
\\end{align}
\\]

Therefore, $(\\simplify{(x+{a})})$ is a factor of $f(x)$.

Similarly for $(\\simplify{(x+{d})})$,

\\[
\\begin{align}
f(\\simplify{-{d}}) &= \\simplify[all,!collectNumbers]{{coef2_x3}+{coef2_x2}+{coef2_x}+{const}}\\\\
&= \\simplify{{coef2_x3}+{coef2_x2}+{coef2_x}+{const}}\\\\
&\\neq 0.
\\end{align}
\\]

Therefore, $(\\simplify{(x+{d})})$ is not a factor of $f(x)$.

Finally, for $(\\simplify{(x+{c})})$,

\\[
\\begin{align}
f(\\simplify{-{c}}) &= \\simplify[all,!collectNumbers]{{coef3_x3}+{coef3_x2}+{coef3_x}+{const}}\\\\
&= \\simplify{{coef3_x3}+{coef3_x2}+{coef3_x}+{const}}.
\\end{align}
\\]

Therefore, $(\\simplify{(x+{c})})$ is also a factor of $f(x)$.

", "statement": "

The factor theorem states that if $f(x)$ is a polynomial and $f(p) = 0$, then $(x-p)$ is a factor of $f(x)$.

", "preamble": {"js": "", "css": ""}, "variables": {"coef1_x3": {"templateType": "anything", "description": "

Number obtained from putting x=-a into the first term of the equation.

", "name": "coef1_x3", "group": "Ungrouped variables", "definition": "(-a)^3"}, "coef1_x2": {"templateType": "anything", "description": "

Number obtained from putting x=-a into the second term of the equation.

", "name": "coef1_x2", "group": "Ungrouped variables", "definition": "(a+b+c)*(-a)^2"}, "coef2_x3": {"templateType": "anything", "description": "

Number obtained from putting x=-d into the first term in the equation.

", "name": "coef2_x3", "group": "Ungrouped variables", "definition": "(-d)^3"}, "coef1_x": {"templateType": "anything", "description": "

Number obtained from putting x=-a into the first term of the equation.

", "name": "coef1_x", "group": "Ungrouped variables", "definition": "(a*b+b*c+a*c)*(-a)"}, "coef3_x3": {"templateType": "anything", "description": "

Number obtained for putting x=-c into the first term of the equation.

", "name": "coef3_x3", "group": "Ungrouped variables", "definition": "(-c)^3"}, "const": {"templateType": "anything", "description": "

Constant term in the equation.

", "name": "const", "group": "Ungrouped variables", "definition": "a*b*c"}, "c": {"templateType": "anything", "description": "

Random number between -2 and 3 except 0 for creating polynomial.

", "name": "c", "group": "Ungrouped variables", "definition": "random(-2..3 except 0)"}, "d": {"templateType": "anything", "description": "

Incorrect answer for part a.

", "name": "d", "group": "Ungrouped variables", "definition": "random(-2..2 except 0 except a except c except b)"}, "coef2_x": {"templateType": "anything", "description": "

Number obtained from putting x=-d into the 3rd term for the equation.

", "name": "coef2_x", "group": "Ungrouped variables", "definition": "(a*b+b*c+a*c)*(-d)"}, "a": {"templateType": "anything", "description": "

Random number between -2 and 3, not including 0 for creating polynomial.

", "name": "a", "group": "Ungrouped variables", "definition": "random(-2..3 except 0 except c)"}, "coef2_x2": {"templateType": "anything", "description": "

Number obtained from putting x=-d into the second term of the equation.

", "name": "coef2_x2", "group": "Ungrouped variables", "definition": "(a+b+c)*(-d)^2"}, "b": {"templateType": "anything", "description": "

Random number between -2 and 3 except 0 for creating polynomial.

", "name": "b", "group": "Ungrouped variables", "definition": "random(-2..3 except 0 except c)"}, "coef3_x": {"templateType": "anything", "description": "

Number obtained by putting x=-c into the third term of the equation.

", "name": "coef3_x", "group": "Ungrouped variables", "definition": "(a*b+b*c+a*c)*(-c)"}, "coef3_x2": {"templateType": "anything", "description": "", "name": "coef3_x2", "group": "Ungrouped variables", "definition": "(a+b+c)*(-c)^2"}}, "parts": [{"variableReplacementStrategy": "originalfirst", "choices": ["

$(\\simplify{x+{a}})$

", "

$(\\simplify{x+{d}})$

", "

$(\\simplify{x+{c}})$

"], "matrix": ["1", 0, "1"], "displayColumns": 0, "distractors": ["", "", ""], "maxMarks": 0, "type": "m_n_2", "minMarks": 0, "marks": 0, "maxAnswers": 0, "prompt": "

Use the factor theorem to find which two of the following are factors of the polynomial

\\[f(x) = \\simplify{x^3+({a}+{b}+{c})x^2+({a}{b}+{a}{c}+{b}{c})x+{a}{b}{c}}.\\]

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Number obtained by putting x=-d into the first term of the equation.

", "group": "Ungrouped variables", "templateType": "anything", "name": "coef_x3", "definition": "(w)*(-d)^3"}, "b": {"description": "

Random number between -2 and 3 except 0 for creating polynomial.

", "group": "Ungrouped variables", "templateType": "anything", "name": "b", "definition": "random(-2..3 except 0)"}, "coef_x": {"description": "

Number obtained by putting x=-d into the third term of the equation.

", "group": "Ungrouped variables", "templateType": "anything", "name": "coef_x", "definition": "(a*d+w*b*d+a*b)*(-d)"}, "d": {"description": "

Used in creation of the polynomial.

", "group": "Ungrouped variables", "templateType": "anything", "name": "d", "definition": "random(-2..2 except 0 except a except b)"}, "w": {"description": "

Random number between 2,3,4.

", "group": "Ungrouped variables", "templateType": "anything", "name": "w", "definition": "random(2,3,4)"}, "coef_x2": {"description": "

Number obtained by putting x=-d into the second term of the equation.

", "group": "Ungrouped variables", "templateType": "anything", "name": "coef_x2", "definition": "(w*d+a+w*b)*(-d)^2"}, "a": {"description": "

Random number between -2 and 3, not including 0 for creating polynomial.

", "group": "Ungrouped variables", "templateType": "anything", "name": "a", "definition": "random(-2..3 except 0)"}}, "functions": {}, "statement": "

The factor theorem states that if $f(x)$ is a polynomial and $f(p) = 0$, then $(x-p)$ is a factor of $f(x)$.

", "variable_groups": [], "parts": [{"scripts": {}, "variableReplacements": [], "marks": 0, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "gaps": [{"checkingtype": "absdiff", "scripts": {}, "type": "jme", "variableReplacementStrategy": "originalfirst", "vsetrange": [0, 1], "showCorrectAnswer": true, "showFeedbackIcon": true, "vsetrangepoints": 5, "checkingaccuracy": 0.001, "variableReplacements": [], "marks": "2", "expectedvariablenames": [], "answer": "{-({w}*({-d})^3+({w}*{d}+{a}+{w}*{b})*({-d})^2+({a}*{d}+{w}*{b}*{d}+{a}*{b})*{-d})}", "checkvariablenames": false, "showpreview": true}], "showFeedbackIcon": true, "prompt": "

Given that $(\\simplify{x+{d}})$ is a factor of $g(x) = \\simplify{{w}*x^3+({w}{d}+{a}+{w}{b})*x^2+({a}{d}+{w}{b}{d}+{a}{b})*x}+m$, find the value of $m$.

$m =$ [[0]].

", "type": "gapfill"}], "ungrouped_variables": ["w", "a", "b", "d", "coef_x3", "coef_x2", "coef_x"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

Given a factor of a cubic polynomial, factorise it fully by first dividing by the given factor, then factorising the remaining quadratic.

"}, "advice": "

Using the factor theorem, we know that if $(x-a)$ is a factor of a polynomial $f(x)$, then $f(a)=0$.

We are given that $(\\simplify{x+{d}})$ is a factor of $g(x) = \\simplify{{w}*x^3+({w}{d}+{a}+{w}{b})*x^2+({a}{d}+{w}{b}{d}+{a}{b})*x+m}$.

By the factor theorem, this means that $g(\\simplify{-{d}}) = 0$.

Substituting $x=\\simplify{-{d}}$ into $g(x)$ gives

\\[
\\begin{align}
g(\\simplify{-{d}}) &= \\simplify[all,!collectNumbers]{{coef_x3}+{coef_x2}+{coef_x}+m}\\\\
&=\\simplify{{coef_x3}+{coef_x2}+{coef_x}+m}.
\\end{align}
\\]

Therefore, as $g(\\simplify{-{d}}) = 0$, we have

\\[
\\begin{align}
\\simplify{{coef_x3}+{coef_x2}+{coef_x}+m}&=0\\\\
m&=\\simplify{-({coef_x3}+{coef_x2}+{coef_x})}.
\\end{align}
\\]

"}, {"name": "Finding the full factorisation of a polynomial, using the Factor Theorem and long division", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Elliott Fletcher", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1591/"}], "type": "question", "statement": "

The factor theorem states that if $f(x)$ is a polynomial and $f(p) = 0$, then $(x-p)$ is a factor of $f(x)$.

", "variablesTest": {"condition": "", "maxRuns": "192"}, "variables": {"y": {"description": "

Factor 3.

", "definition": "random(-2..3 except 0)", "templateType": "anything", "name": "y", "group": "Ungrouped variables"}, "u": {"description": "

Factor 2.

", "definition": "random(-2..3 except 0 except -y)", "templateType": "anything", "name": "u", "group": "Ungrouped variables"}, "z": {"description": "

Factor 1.

", "definition": "random(-3..3 except 0)", "templateType": "anything", "name": "z", "group": "Ungrouped variables"}}, "functions": {}, "tags": ["factor theorem", "Factor Theorem", "factorisation", "Factorisation", "factorise", "Factorise", "Long division", "long division", "polynomials", "Polynomials", "taxonomy"], "variable_groups": [], "parts": [{"extendBaseMarkingAlgorithm": true, "vsetRange": [0, 1], "scripts": {}, "type": "jme", "variableReplacementStrategy": "originalfirst", "checkingType": "absdiff", "showFeedbackIcon": true, "prompt": "

Given that $(\\simplify{x+{z}})$ is a factor of \\[p(x) = \\simplify{x^3+({y}+{u}+{z})*x^2+({y}*{u}+{z}*{u}+{y}*{z})*x+{y}*{u}*{z}}.\\]

Find the full factorisation of $p(x)$.

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Use a given factor of a polynomial to find the full factorisation of the polynomial through long division.

"}, "preamble": {"css": "", "js": ""}, "advice": "

For this question, we are given that $(\\simplify{x+{z}})$ is a factor of the polynomial

\\[p(x) = \\simplify{x^3+({y}+{u}+{z})*x^2+({y}{u}+{z}{u}+{y}{z})*x+{y}{u}{z}},\\]

and we are then asked to find the full factorisation of $p(x)$.

We know that $(\\simplify{x+{z}})$ is a factor of $p(x)$, so we can calculate the other factors of $p(x)$ through long division.

\\[
\\begin{align}
&\\simplify{x^2+({u}+{y})x+{u}{y}}\\\\
\\simplify{x+{z}} \\; &\\overline{)\\simplify{x^3+({y}+{u}+{z})*x^2+({y}{u}+{z}{u}+{y}{z})*x+{y}{u}{z}}}\\\\
&\\;\\,
\\simplify{x^3+{z}x^2}\\\\
&\\qquad\\quad
\\overline{\\simplify[all,noLeadingMinus]{({u}+{y})x^2+({y}{u}+{z}{u}+{y}{z})*x+{y}{u}{z}}}\\\\
&\\qquad\\quad
\\simplify[all,noLeadingMinus]{({u}+{y})x^2+({u}{z}+{z}{y})x}\\\\
&\\qquad\\quad\\quad\\quad\\quad
\\overline{\\simplify[all,noLeadingMinus]{{y}{u}x+{y}{u}{z}}}\\\\
&\\qquad\\quad\\quad\\quad\\quad
\\simplify[all,noLeadingMinus]{{y}{u}x+{y}{u}{z}}\\\\
&\\qquad\\qquad\\quad\\quad\\quad
\\overline{0.}
\\end{align}
\\]

We can then factorise $\\simplify{x^2+({u}+{y})x+{u}{y}}$ into

\\[\\simplify{x^2+({u}+{y})x+{u}{y}} =(\\simplify{x+{y}})(\\simplify{x+{u}}).\\]

Therefore, the full factorisation of $p(x)$ is

\\[
\\begin{align}
p(x) &= \\simplify{x^3+({y}+{u}+{z})*x^2+({y}{u}+{z}{u}+{y}{z})*x+{y}{u}{z}},\\\\
&= (\\simplify{x+{y}})(\\simplify{x+{z}})(\\simplify{x+{u}}).
\\end{align}
\\]

"}, {"name": "Finding unknown coefficients of a polynomial, using the remainder theorem", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Elliott Fletcher", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1591/"}], "type": "question", "tags": ["taxonomy"], "variablesTest": {"condition": "x > y", "maxRuns": 100}, "variables": {"numerator": {"definition": "(rem1 - rem2-coef2_x3*(-c)^3-coef2_x*(-c)+coef2_x3*(-d)^3+coef2_x*(-d))*(-d)^2", "name": "numerator", "description": "

Numerator of s

", "templateType": "anything", "group": "Ungrouped variables"}, "c": {"definition": "random(-3..3 except [0,1])", "name": "c", "description": "

Dividing term 1.

", "templateType": "anything", "group": "Ungrouped variables"}, "rem1": {"definition": "random(1..3)", "name": "rem1", "description": "

First remainder.

", "templateType": "anything", "group": "Ungrouped variables"}, "s": {"definition": "numerator/denominator", "name": "s", "description": "", "templateType": "anything", "group": "Ungrouped variables"}, "coef2_x": {"definition": "random(-3..3 except [0,c,d])", "name": "coef2_x", "description": "

Coefficient of x.

", "templateType": "anything", "group": "Ungrouped variables"}, "denominator": {"definition": "(-c)^2-(-d)^2", "name": "denominator", "description": "

Denominator of s.

", "templateType": "anything", "group": "Ungrouped variables"}, "t": {"definition": "(rem2-coef2_x3*(-d)^3-coef2_x*(-d) - (rem1 - rem2-coef2_x3*(-c)^3-coef2_x*(-c)+coef2_x3*(-d)^3+coef2_x*(-d))/((-c)^2-(-d)^2)*(-d)^2)", "name": "t", "description": "", "templateType": "anything", "group": "Ungrouped variables"}, "d": {"definition": "random(-3..3 except c except [0,1,-c])", "name": "d", "description": "

Dividing term 2.

", "templateType": "anything", "group": "Ungrouped variables"}, "coef2_x3": {"definition": "random(1..2)", "name": "coef2_x3", "description": "

Coefficient of x^3.

", "templateType": "anything", "group": "Ungrouped variables"}, "x": {"definition": "(-(c))^2", "name": "x", "description": "

Simplifies first coefficient of s.

", "templateType": "anything", "group": "Ungrouped variables"}, "y": {"definition": "(-(d))^2", "name": "y", "description": "

Simplifies second coefficient of s.

", "templateType": "anything", "group": "Ungrouped variables"}, "rem2": {"definition": "random(-3..3 except 0) ", "name": "rem2", "description": "

Second remainder.

", "templateType": "anything", "group": "Ungrouped variables"}}, "statement": "

Consider the polynomial

\\[ p(x) = \\simplify{{coef2_x3}x^3+s*x^2+{coef2_x}x+t}\\text{.}\\]

The polynomial:

has remainder $\\var{rem1}$ when divided by $(\\simplify{x+{c}})$
has remainder $\\var{rem2}$ when divided by $(\\simplify{x+{d}})$

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Using the remainder theorem for the remainder when $p(x)$ is divided by $(\\simplify{x+{c}})$, create an equation involving $s$ and $t$.

[[0]]$s + t$ = [[1]].

", "steps": [{"extendBaseMarkingAlgorithm": true, "scripts": {}, "type": "information", "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "showFeedbackIcon": true, "prompt": "

The remainder theorem states that if a polynomial $f(x)$ is divided by $(\\simplify{a*x-b})$ then the remainder is $f(\\frac{b}{a})$.

", "useCustomName": false, "variableReplacements": [], "marks": 0, "customName": "", "showCorrectAnswer": true, "unitTests": []}], "useCustomName": false, "variableReplacements": [], "marks": 0, "customName": "", "showCorrectAnswer": true, "unitTests": []}, {"extendBaseMarkingAlgorithm": true, "scripts": {}, "type": "gapfill", "variableReplacementStrategy": "originalfirst", "sortAnswers": false, "customMarkingAlgorithm": "", "gaps": [{"correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "notationStyles": ["plain", "en", "si-en"], "type": "numberentry", "variableReplacementStrategy": "originalfirst", "showFractionHint": true, "customMarkingAlgorithm": "", "maxValue": "{d}^2", "showFeedbackIcon": true, "allowFractions": false, "correctAnswerStyle": "plain", "useCustomName": false, "mustBeReduced": false, "minValue": "{d}^2", "variableReplacements": [], "marks": 1, "customName": "", "scripts": {}, "showCorrectAnswer": true, "unitTests": [], "mustBeReducedPC": 0}, {"correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "notationStyles": ["plain", "en", "si-en"], "type": "numberentry", "variableReplacementStrategy": "originalfirst", "showFractionHint": true, "customMarkingAlgorithm": "", "maxValue": "{rem2}+{coef2_x3}*(d^3)+{coef2_x}*d", "showFeedbackIcon": true, "allowFractions": false, "correctAnswerStyle": "plain", "useCustomName": false, "mustBeReduced": false, "minValue": "{rem2}+{coef2_x3}*(d^3)+{coef2_x}*d", "variableReplacements": [], "marks": 1, "customName": "", "scripts": {}, "showCorrectAnswer": true, "unitTests": [], "mustBeReducedPC": 0}], "showFeedbackIcon": true, "prompt": "

Using the remainder theorem for the remainder when $p(x)$ is divided by $(\\simplify{x+{d}})$, create another equation involving $s$ and $t$.

[[0]]$s+t$ = [[1]].

", "useCustomName": false, "variableReplacements": [], "marks": 0, "customName": "", "showCorrectAnswer": true, "unitTests": []}, {"extendBaseMarkingAlgorithm": true, "scripts": {}, "stepsPenalty": 0, "type": "gapfill", "variableReplacementStrategy": "originalfirst", "sortAnswers": false, "customMarkingAlgorithm": "", "gaps": [{"correctAnswerFraction": true, "extendBaseMarkingAlgorithm": true, "notationStyles": ["plain", "en", "si-en"], "type": "numberentry", "variableReplacementStrategy": "originalfirst", "showFractionHint": true, "customMarkingAlgorithm": "", "maxValue": "s", "showFeedbackIcon": true, "allowFractions": true, "correctAnswerStyle": "plain", "useCustomName": false, "mustBeReduced": true, "minValue": "s", "variableReplacements": [], "marks": 1, "customName": "", "scripts": {}, "showCorrectAnswer": true, "unitTests": [], "mustBeReducedPC": 0}], "showFeedbackIcon": true, "prompt": "

Find the value of $s$. Reduce your answer to its simplest fractional form.

$s =$ [[0]]

Subtract the two simultaneous equations for $s$ and $t$, obtained in parts a) and b), from each other.

Then rearrange this new equation to find the value of $s$.

", "useCustomName": false, "variableReplacements": [], "marks": 0, "customName": "", "showCorrectAnswer": true, "unitTests": []}], "useCustomName": false, "variableReplacements": [], "marks": 0, "customName": "", "showCorrectAnswer": true, "unitTests": []}, {"extendBaseMarkingAlgorithm": true, "scripts": {}, "stepsPenalty": 0, "type": "gapfill", "variableReplacementStrategy": "originalfirst", "sortAnswers": false, "customMarkingAlgorithm": "", "gaps": [{"correctAnswerFraction": true, "extendBaseMarkingAlgorithm": true, "notationStyles": ["plain", "en", "si-en"], "type": "numberentry", "variableReplacementStrategy": "originalfirst", "showFractionHint": true, "customMarkingAlgorithm": "", "maxValue": "{(rem2-coef2_x3*(-d)^3-coef2_x*(-d) - (rem1 - rem2-coef2_x3*(-c)^3-coef2_x*(-c)+coef2_x3*(-d)^3+coef2_x*(-d))/((-c)^2-(-d)^2)*(-d)^2)}", "showFeedbackIcon": true, "allowFractions": true, "correctAnswerStyle": "plain", "useCustomName": false, "mustBeReduced": true, "minValue": "{(rem2-coef2_x3*(-d)^3-coef2_x*(-d) - (rem1 - rem2-coef2_x3*(-c)^3-coef2_x*(-c)+coef2_x3*(-d)^3+coef2_x*(-d))/((-c)^2-(-d)^2)*(-d)^2)}", "variableReplacements": [], "marks": 1, "customName": "", "scripts": {}, "showCorrectAnswer": true, "unitTests": [], "mustBeReducedPC": 0}], "showFeedbackIcon": true, "prompt": "

Find the value of $t$. Reduce your answer to its simplest fractional form.

$t =$ [[0]]

Substitute the value of $s$ from part c) into one of the simultaneous equations for $s$ and $t$.

Then, rearrange this equation to find the value of $t$.

This question tests the student's knowledge of the remainder theorem and the ways in which it can be applied.

"}, "preamble": {"css": "", "js": ""}, "functions": {}, "advice": "

We are told that the polynomial:

has a remainder $\\var{rem1}$ when divided by $(\\simplify{x+{c}})$
has a remainder $\\var{rem2}$ when divided by $(\\simplify{x+{d}})$

a)

Firstly, substituting $x = \\simplify{-{c}}$ into $p(x)$ gives us

\\begin{align}
p(\\simplify{-{c}}) &= \\simplify[all,!collectNumbers, fractionnumbers]{{coef2_x3*(-c)^3}+{(-c)^2}s+{coef2_x*(-c)}+t},\\\\
&= \\simplify[all,fractionnumbers]{{coef2_x3*(-c)^3}+{(-c)^2}s+{coef2_x*(-c)}+t}.
\\end{align}

But, by the remainder theorem $p(\\simplify{-{c}}) = \\var{rem1}$ (using the first bullet point), so this becomes

\\begin{align}
\\simplify[all,fractionnumbers]{{coef2_x3*(-{c})^3}+s*{(-{c})^2}+{coef2_x*(-{c})}+t} &= \\var{rem1},\\\\
\\simplify[all,fractionnumbers]{s*{x}+t} &= \\simplify[all,fractionnumbers]{{rem1}-{coef2_x3*(-{c})^3}-{coef2_x*(-{c})}}.
\\end{align}

b)

Similarly, substituting $x = \\simplify{-{d}}$ into $p(x)$, gives us

\\begin{align}
p(\\simplify{-{d}}) &= \\simplify[all,!collectNumbers, fractionnumbers]{{coef2_x3*(-{d})^3}+{(-{d})^2}s+{coef2_x*(-{d})}+t},\\\\
&= \\simplify[all,fractionnumbers]{{coef2_x3*(-{d})^3}+{(-{d})^2}s+{coef2_x*(-{d})}+t}.
\\end{align}

But, by the remainder theorem $p(\\simplify{-{d}}) = \\var{rem2}$ (using the second bullet point), so this becomes

\\begin{align}
\\simplify[all,fractionnumbers]{{coef2_x3*(-{d})^3}+s*{(-{d})^2}+{coef2_x*(-{d})}+t} &= \\var{rem2},\\\\
\\simplify[all,fractionnumbers]{s*{y}+t} &= \\simplify[all,fractionnumbers]{{rem2}-{coef2_x3*(-{d})^3}-{coef2_x*(-{d})}}.
\\end{align}

c)

We now have two simultaneous equations for $s$ and $t$:

\\begin{align}
\\simplify[all,fractionnumbers]{s*{x}+t} = \\simplify[all,fractionnumbers]{{rem1}-{coef2_x3*(-{c})^3}-{coef2_x*(-{c})}} \\\\
\\simplify[all,fractionnumbers]{s*{y}+t} = \\simplify[all,fractionnumbers]{{rem2}-{coef2_x3*(-{d})^3}-{coef2_x*(-{d})}}
\\end{align}

Next, we subtract the second equation from the first equation.

This allows us to cancel out the terms involving $t$ and gives us an equation only in terms of $s$, which we can then rearrange to find the value of $s$.

Subtracting the two equations gives

\\[\\simplify{s*{(-{c})^2-(-{d})^2}} = \\simplify[all,fractionnumbers]{{rem1 - coef2_x3*(-c)^3-coef2_x*(-c)-rem2+coef2_x3*(-d)^3+coef2_x*(-d)}}.\\]

Then, we can rearrange this equation so that

\\[s = \\simplify[all,fractionnumbers]{{rem1 - coef2_x3*(-c)^3-coef2_x*(-c)-rem2+coef2_x3*(-d)^3+coef2_x*(-d)}/{{(-c)^2-(-d)^2}}}.\\]

d)

We can calculate $t$ by substituting our value of $s$ into one of our original simultaneous equations. For example, let's use the equation

\\[\\simplify[all,fractionnumbers]{s*{(-{d})^2}+t} = \\simplify[all,fractionnumbers]{{rem2}-{coef2_x3*(-{d})^3}-{coef2_x*(-{d})}}.\\]

Substituting our value of $s$ into this equation gives us

\\[
\\begin{align}
\\simplify[all,fractionnumbers,!noleadingMinus]{{numerator/denominator}+t} &= \\simplify[all,fractionnumbers]{{rem2-coef2_x3*(-d)^3-coef2_x*(-d)}},\\\\
t &= \\simplify[all,fractionnumbers]{{rem2-coef2_x3*(-d)^3-coef2_x*(-d) - numerator/denominator}}.
\\end{align}
\\]

This same answer would've also been obtained if we had substituted our value of $s$ into the other equation instead.

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Constant term

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Coefficient of x^2.

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Coefficient of x.

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Coefficient of x^3

", "templateType": "anything", "definition": "random(2..4) "}, "remainder": {"group": "Ungrouped variables", "name": "remainder", "description": "

Correct remainder.

", "templateType": "anything", "definition": "coef_x3*(-k/a)^3+coef_x2*(-k/a)^2+coef_x*(-k/a)+const"}, "k": {"group": "Ungrouped variables", "name": "k", "description": "

Free coefficient in the dividing equation.

", "templateType": "anything", "definition": "random(-3..3 except 0 except 1 except -1)"}, "a": {"group": "Ungrouped variables", "name": "a", "description": "

Leading coefficient in the dividing equation.

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Find the remainder when $f(x) = \\simplify{{coef_x3}x^3+{coef_x2}x^2+{coef_x}x+{const}}$ is divided by $(\\simplify{{a}x+{k}})$, using the remainder theorem.

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This question tests the student's ability to find remainders using the remainder theorem.

"}, "preamble": {"css": "", "js": ""}, "functions": {}, "advice": "

The remainder theorem states that if a polynomial $f(x)$ is divided by $(\\simplify{a*x-b})$ then the remainder is $f \\left( \\frac{b}{a} \\right)$.

This means that if we substitute $x = \\frac{b}{a}$ into the equation for $f(x)$, the result will be equal to the remainder when $f(x)$ is divided by $(\\simplify{a*x-b})$.

Therefore, to calculate the remainder when $f(x) = \\simplify{{coef_x3}*x^3+{coef_x2}*x^2+{coef_x}*x+{const}}$ is divided by $(\\simplify{{a}*x+{k}})$, we use this same principle.

As we are dividing $f(x)$ by $(\\simplify{{a}*x+{k}})$, using the remainder theorem tells us that substituting

\\[
\\begin{align}
x &= \\frac{b}{a}\\\\
&= \\simplify{-({k}/{a})}
\\end{align}
\\]

into our equation for $f(x)$ will give us the remainder when $f(x)$ is divided by $(\\simplify{{a}*x+{k}})$. Substituting this value of $x$ into $f(x)$ gives us

\\[
\\begin{align}
f(\\simplify{-({k}/{a})}) &= \\simplify[all,!collectNumbers, fractionnumbers]{{coef_x3*(-({k}/{a}))^3}+{coef_x2*(-({k}/{a}))^2}+{coef_x*(-({k}/{a}))}+{const}}\\\\
&= \\simplify[all,fractionnumbers]{{coef_x3*(-({k}/{a}))^3}+{coef_x2*(-({k}/{a}))^2}+{coef_x*(-({k}/{a}))}+{const}}.
\\end{align}
\\]

Therefore, the remainder when $f(x) = \\simplify{{coef_x3}*x^3+{coef_x2}*x^2+{coef_x}*x+{const}}$ is divided by $(\\simplify{{a}*x+{k}})$ is $\\displaystyle\\simplify[all,fractionnumbers]{{coef_x3*(-({k}/{a}))^3}+{coef_x2*(-({k}/{a}))^2}+{coef_x*(-({k}/{a}))}+{const}}$.

"}, {"name": "Inverse and composite functions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Bradley Bush", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1521/"}], "rulesets": {}, "functions": {}, "ungrouped_variables": ["a", "b"], "metadata": {"description": "

Find the inverse of a composite function by finding the inverses of two functions and then the composite of these; and by finding the composite of two functions then finding the inverse. The question then concludes by asking students to compare their two answers and verify they're equivalent.

", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "variable_groups": [], "advice": "

a)

$f^{-1}(x)$ is the function with the property that $f^{-1}(f(x)) = x$.

To find this, we first set $x=f(y)$ and rearrange to find $y$ in terms of $x$, i.e. $y = f^{-1}(x)$.

\\begin{align}
f(y)=\\simplify{{a[0]}y+{a[1]}}&=x\\\\
\\simplify{{a[0]}y}&=x-\\var{a[1]}\\\\[1em]
y&=\\simplify[]{(x-{a[1]})/{a[0]}}\\\\[1em]
f^{-1}(x)&=\\simplify{(x-{a[1]})/{a[0]}}\\text{.}\\\\
\\end{align}

b)

We use the same method as part a) to find $g^{-1}(x)$:

\\begin{align}
g(y)=\\simplify{{a[3]}y-{a[2]}}&=x\\\\
\\simplify{{a[3]}y}&=x+\\var{a[2]}\\\\[1em]
y&=\\simplify[]{(x+{a[2]})/{a[3]}}\\\\[1em]
g^{-1}(x)&=\\simplify{(x+{a[2]})/{a[3]}}\\text{.}\\\\
\\end{align}

c)

$(g^{-1} \\circ f^{-1})(x)$ is the function which first applies $f^{-1}(x)$ and then applies $g^{-1}$ to the result of that.

We use the previous answers: $f^{-1}(x)=\\simplify{(x-{a[1]})/{a[0]}}$ and $g^{-1}(x)=\\simplify{(x+{a[2]})/{a[3]}}$ to find the definition of $(g^{-1} \\circ f^{-1})(x)$ by substituting $f^{-1}(x)$ everywhere $x$ occurs in the definition of $g^{-1}(x)$.

\\begin{align}
(g^{-1}\\circ f^{-1}) (x)&=g^{-1}(f^{-1}(x))\\\\[1em]
&=g^{-1} \\left( \\simplify[]{(x-{a[1]})/{a[0]}} \\right) \\\\[1em]
&=\\frac{\\left(\\simplify[]{(x-{a[1]})/{a[0]}}\\right)+\\simplify[]{{a[2]}}}{\\var{a[3]}\\text{.}}
\\end{align}

d)

$(f \\circ g)(x)$ is the function which first applies $g(x)$ and then applies $f$ to the result of that.

We find the definition of $(f \\circ g)(x)$ by substituting $g(x)$ everywhere that $x$ occurs in the definition of $f(x)$.

\\begin{align}
(f\\circ g)(x)&=f(g(x))\\\\
&=f(\\simplify{{a[3]}x-{a[2]}})\\\\
&=\\simplify{{a[0]}({a[3]}x-{a[2]})+{a[1]}}
\\end{align}

e)

Now that we have the definition of $(f \\circ g)(x)$, we can find its inverse by using the same method as in parts a) and b).

\\begin{align}
(f \\circ g)(y) &= x \\\\
\\simplify{{a[0]}({a[3]}y-{a[2]})+{a[1]}}&=x\\\\
\\simplify{{a[0]}({a[3]}y-{a[2]})}&=x-\\var{a[1]}\\\\[1em]
\\simplify{{a[3]}y-{a[2]}}&=\\frac{(x-\\var{a[1]})}{\\var{a[0]}}\\\\[1em]
\\simplify{{a[3]}y}&=\\left( \\frac{(x-\\var{a[1]})}{\\var{a[0]}} \\right) +\\var{a[2]}\\\\[1em]
y&=\\frac{\\left(\\frac{(x-\\var{a[1]})}{\\var{a[0]}}\\right)+\\var{a[2]})}{\\var{a[3]}}\\\\[1em]
(f\\circ g)^{-1}(x)&=\\frac{\\left(\\frac{(x-\\var{a[1]})}{\\var{a[0]}}\\right)+\\var{a[2]}}{\\var{a[3]}}\\\\[1em]
\\end{align}

We can see that in this case $(f\\circ g)^{-1}(x) = (g^{-1}\\circ f^{-1}) (x)$.

f)

So long as the inverses of $f$ and $g$ exist and they can be composed, it is always true that \\[(f \\circ g)^{-1}(x) \\equiv (g^{-1} \\circ f^{-1}) (x)\\text{.}\\]

", "statement": "

Given a function $f(x)$, the inverse function $f^{-1}(x)$ reverses whatever $f$ does.

Function composition is applying one function to the results of another.

The following questions will ask you to find the inverses and compositions of some functions.

Give all of your answers in terms of $x$.

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Find $f^{-1}(x)$ when $\\simplify{f(x)={a[0]}x+{a[1]} }$.

$\\displaystyle f^{-1}(x)=$ [[0]]

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Find $g^{-1}(x)$ when $\\simplify{g(x)={a[3]}x-{a[2]} }$.

$\\displaystyle g^{-1}(x)=$ [[0]]

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Use your above answers for $f^{-1}(x)$ and $g^{-1}(x)$ to find the inverse, composed function, $(g^{-1}\\circ f^{-1}) (x)$ terms of $x$:
$\\displaystyle (g^{-1}\\circ f^{-1}) (x)$ =[[0]]

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Using:
\\[
\\begin{align}
f(x)&=\\simplify{{a[0]}x+{a[1]} }\\\\
&\\text{ and } \\\\
g(x)&=\\simplify{{a[3]}x-{a[2]} }\\text{,}
\\end{align}
\\]
find $(f\\circ g)(x)$, the composition of $f(x)$ with $g(x)$.

$\\displaystyle (f\\circ g)(x)=$ [[0]]

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Use your above answer for $(f\\circ g)(x)$ to find the inverse, composed function,$(f\\circ g)^{-1}(x)$ in terms of $x$:

$\\displaystyle (f\\circ g)^{-1}(x)=$ [[0]]

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Never

", "

Only when $f(x)=g(x)$

", "

Only when $f(x)$ is in the same family as $g(x)$

", "

Always, provided that composite and inverse functions exist

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When should your answer for c), $(g^{-1}\\circ f^{-1}) (x)$ be the same as your answer for e) $(f\\circ g)^{-1}(x)$?

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