Apply the factor and remainder theorems to manipulate polynomial expressions
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", "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.
\nIf $f(x)$ is a polynomial and $f(p) = 0$, then $(x-p)$ is a factor of $f(x)$.
\nIf $(\\simplify{(x+{a})})$ is a factor of $f(x)$ then by the factor theorem, $f(\\simplify{-{a}}) = 0$.
\nWe see that
\n\\[
\\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 $f(x)$.
\nSimilarly for $(\\simplify{(x+{d})})$,
\n\\[
\\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 $f(x)$.
\nFinally, for $(\\simplify{(x+{c})})$,
\n\\[
\\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 $f(x)$.
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", "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": ["$(\\simplify{x+{a}})$
", "$(\\simplify{x+{d}})$
", "$(\\simplify{x+{c}})$
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\n\\[f(x) = \\simplify{x^3+({a}+{b}+{c})x^2+({a}{b}+{a}{c}+{b}{c})x+{a}{b}{c}}.\\]
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", "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 $m$.
\n$m =$ [[0]].
\n", "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 $f(a)=0$.
\nWe are given that $(\\simplify{x+{d}})$ is a factor of $g(x) = \\simplify{{w}*x^3+({w}{d}+{a}+{w}{b})*x^2+({a}{d}+{w}{b}{d}+{a}{b})*x+m}$.
\nBy the factor theorem, this means that $g(\\simplify{-{d}}) = 0$.
\nSubstituting $x=\\simplify{-{d}}$ into $g(x)$ gives
\n\\[
\\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
\n\\[
\\begin{align}
\\simplify{{coef_x3}+{coef_x2}+{coef_x}+m}&=0\\\\
m&=\\simplify{-({coef_x3}+{coef_x2}+{coef_x})}.
\\end{align}
\\]
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}}.\\]
\nFind the full factorisation of $p(x)$.
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"}, "preamble": {"css": "", "js": ""}, "advice": "For this question, we are given that $(\\simplify{x+{z}})$ is a factor of the polynomial
\n\\[p(x) = \\simplify{x^3+({y}+{u}+{z})*x^2+({y}{u}+{z}{u}+{y}{z})*x+{y}{u}{z}},\\]
\nand we are then asked to find the full factorisation of $p(x)$.
\nWe know that $(\\simplify{x+{z}})$ is a factor of $p(x)$, so we can calculate the other factors of $p(x)$ through long division.
\n\\[
\\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
\n\\[\\simplify{x^2+({u}+{y})x+{u}{y}} =(\\simplify{x+{y}})(\\simplify{x+{u}}).\\]
\nTherefore, the full factorisation of $p(x)$ is
\n\\[
\\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}
\\]
Numerator of s
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", "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
\n\\[ p(x) = \\simplify{{coef2_x3}x^3+s*x^2+{coef2_x}x+t}\\text{.}\\]
\nThe polynomial:
\nUsing the remainder theorem for the remainder when $p(x)$ is divided by $(\\simplify{x+{c}})$, create an equation involving $s$ and $t$.
\n[[0]]$s + t$ = [[1]].
\n", "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 $t$.
\n[[0]]$s+t$ = [[1]].
\n", "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.
\n$s =$ [[0]]
", "steps": [{"extendBaseMarkingAlgorithm": true, "scripts": {}, "type": "information", "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "showFeedbackIcon": true, "prompt": "Subtract the two simultaneous equations for $s$ and $t$, obtained in parts a) and b), from each other.
\nThen 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.
\n$t =$ [[0]]
", "steps": [{"extendBaseMarkingAlgorithm": true, "scripts": {}, "type": "information", "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "showFeedbackIcon": true, "prompt": "Substitute the value of $s$ from part c) into one of the simultaneous equations for $s$ and $t$.
\nThen, rearrange this equation to find the value of $t$.
", "useCustomName": false, "variableReplacements": [], "marks": 0, "customName": "", "showCorrectAnswer": true, "unitTests": []}], "useCustomName": false, "variableReplacements": [], "marks": 0, "customName": "", "showCorrectAnswer": true, "unitTests": []}], "ungrouped_variables": ["c", "rem1", "d", "rem2", "coef2_x3", "coef2_x", "numerator", "denominator", "s", "x", "y", "t"], "rulesets": {}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "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:
\nFirstly, substituting $x = \\simplify{-{c}}$ into $p(x)$ gives us
\n\\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
\n\\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}
Similarly, substituting $x = \\simplify{-{d}}$ into $p(x)$, gives us
\n\\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
\n\\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}
We now have two simultaneous equations for $s$ and $t$:
\n\\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.
\nThis 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 $s$.
\nSubtracting the two equations gives
\n\\[\\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)}}.\\]
\nThen, we can rearrange this equation so that
\n\\[s = \\simplify[all,fractionnumbers]{{rem1 - coef2_x3*(-c)^3-coef2_x*(-c)-rem2+coef2_x3*(-d)^3+coef2_x*(-d)}/{{(-c)^2-(-d)^2}}}.\\]
\nWe can calculate $t$ by substituting our value of $s$ into one of our original simultaneous equations. For example, let's use the equation
\n\\[\\simplify[all,fractionnumbers]{s*{(-{d})^2}+t} = \\simplify[all,fractionnumbers]{{rem2}-{coef2_x3*(-{d})^3}-{coef2_x*(-{d})}}.\\]
\nSubstituting our value of $s$ into this equation gives us
\n\\[
\\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.
"}, {"name": "Dividing a polynomial with remainders, 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", "statement": "", "variablesTest": {"condition": "", "maxRuns": 100}, "variables": {"const": {"group": "Ungrouped variables", "name": "const", "description": "Constant term
", "templateType": "anything", "definition": "random(-5..5 except 0)"}, "coef_x2": {"group": "Ungrouped variables", "name": "coef_x2", "description": "Coefficient of x^2.
", "templateType": "anything", "definition": "random(-3..3 except 0)"}, "coef_x": {"group": "Ungrouped variables", "name": "coef_x", "description": "Coefficient of x.
", "templateType": "anything", "definition": "random(-3..3 except 0)"}, "coef_x3": {"group": "Ungrouped variables", "name": "coef_x3", "description": "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.
", "templateType": "anything", "definition": "random(1..3 except k except -k)"}}, "tags": ["taxonomy"], "variable_groups": [], "parts": [{"correctAnswerFraction": true, "notationStyles": ["plain", "en", "si-en"], "type": "numberentry", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "maxValue": "remainder", "showFeedbackIcon": true, "prompt": "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.
", "correctAnswerStyle": "plain", "allowFractions": true, "mustBeReduced": true, "minValue": "remainder", "variableReplacements": [], "marks": "2", "scripts": {}, "showCorrectAnswer": true}], "ungrouped_variables": ["coef_x3", "coef_x2", "coef_x", "const", "k", "remainder", "a"], "rulesets": {}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "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 $f \\left( \\frac{b}{a} \\right)$.
\nThis 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 $(\\simplify{a*x-b})$.
\nTherefore, 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.
\nAs we are dividing $f(x)$ by $(\\simplify{{a}*x+{k}})$, using the remainder theorem tells us that substituting
\n\\[
\\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
\n\\[
\\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 $\\displaystyle\\simplify[all,fractionnumbers]{{coef_x3*(-({k}/{a}))^3}+{coef_x2*(-({k}/{a}))^2}+{coef_x*(-({k}/{a}))}+{const}}$.
"}]}], "percentPass": 0, "duration": 0, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}], "extensions": [], "custom_part_types": [], "resources": []}