mathcentre: Combining algebraic fractions

Number of Questions:	13
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Question 1

Add the following together and express as a single algebraic fraction.

Express $\simplify[std]{{a} + ({c} / ({a2}x + {d}))}$ as a single algebraic fraction.

Input the fraction here: $\displaystyle{}$ Expected answer: .

Input your answer in the form $\displaystyle \frac{(ax+b)}{(cx+d)}$ with no other brackets than those shown.

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The formula for adding these expressions is :
$\simplify[std]{a + {s1} * (c / d) = (ad + {s1} * bc) / d}$

and for this exercise we have $\simplify{d={a2}x+{d}}$ .

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We have:

$\simplify[std]{{a} + ({c} / ({a2}*x + {d})) = ({a} * ({a2}*x + {d}) + {c}) / (({a2}*x + {d})) = ({a*a2} * x + {a * d + c}) / ( ({a2}*x + {d}))}$

$\endgroup$

Question 2

Add the following two fractions together and express as a single fraction over a common denominator.

Express

\[\simplify{{a} / ({a1}x + {b}) + ({c} / ({a2}x + {d}))}\]

as a single fraction.

Enter the fraction here: $\displaystyle{}$ Expected answer:

Input your answer in the form $\displaystyle \frac{(ax+b)}{((cx+d)(ex+f))}$ with no other brackets than those shown.

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Advice

The formula for taking away fractions is :

\[\simplify[std]{a / b + {s1} * (c / d) = (ad + {s1} * bc) / bd}\]

and for this exercise we have $\simplify{b=x+{b}}$, $\simplify{d=x+{d}}$.
Hence we have:
\[\simplify[std]{{a} / ({a1}*x + {b}) + ({c} / ({a2}*x + {d})) = ({a} * ({a2}*x + {d}) + {c} * ({a1}*x + {b})) / (({a1}*x + {b}) * ({a2}*x + {d})) = ({a*a2 + c*a1} * x + {a * d + c * b}) / (({a1}*x + {b}) * ({a2}*x + {d}))}\]

Question 3

Express the following as a single fraction.

Express \[\simplify[std]{{a}x+{b1} } +\simplify[std]{ ({c}x+{b2}) / ({a2}x + {d})}\] as a single fraction.

Input the fraction here: $\displaystyle{}$ Expected answer: .

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Advice

The formula for adding these expressions is:

\[\simplify[std]{a + (c / d) = (ad + c) / d}\]

and for this exercise we have $\simplify{a={b1}}$, $\simplify{c={c}x+{b2}}$, $\simplify{d={a2}x+{d}}$.

Hence we have:
\[\begin{eqnarray*} \simplify[std]{{b1} } +\simplify[std]{ ({c}x+{b2}) / ({a2}x + {d})} &=& \simplify[basic,unitFactor]{(({b1}) * ({a2}*x + {d}) + ({c}x+{b2}) ) / ( ({a2}*x + {d}))}\\ &=&\simplify[std]{ (({b1*a2}x+{b1*d})+{c}x+{b2}) / ( ({a2}*x + {d}))}\\&=&\simplify[std]{ ( {b1*a2+ c }x+{b1*d+b2}) / (({a2}*x + {d}))}\end{eqnarray*}\]

Question 4

Express the following as a single fraction.

Express \[\simplify[std]{{a}x+{b1} } +\simplify[std]{ ({c}x+{b2}) / ({a2}x + {d})}\] as a single fraction.

Input the fraction here: $\displaystyle{}$ Expected answer: .

Make sure that you simplify the numerator.

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Advice

The formula for adding these expressions is:

\[\simplify[std]{a + (c / d) = (ad + c) / d}\]

and for this exercise we have $\simplify{a={a}x+{b1}}$, $\simplify{c={c}x+{b2}}$, $\simplify{d={a2}x+{d}}$.

Hence we have:
\[\begin{eqnarray*} \simplify[std]{{a}x+{b1} } +\simplify[std]{ ({c}x+{b2}) / ({a2}x + {d})} &=& \simplify{(({a}x+{b1}) * ({a2}*x + {d}) + ({c}x+{b2}) ) / ( ({a2}*x + {d}))}\\ &=&\simplify[std]{ (({a*a2} * x^2 + {b1*a2+ a*d}x+{b1*d})+{c}x+{b2}) / ( ({a2}*x + {d}))}\\&=&\simplify[std]{ ({a*a2} * x^2 + {a * d +b1*a2+ c }x+{b1*d+b2}) / (({a2}*x + {d}))}\end{eqnarray*}\]

Question 5

Add the following two fractions together and express as a single fraction over a common denominator.

Express \[\simplify{({a}x+{b1}) / ({a1}x + {b}) + ({c}x+{b2}) / ({a2}x + {d})}\] as a single fraction.

Input the fraction here: $\displaystyle{}$ Expected answer: .

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Advice

The formula for taking away fractions is :

\[\simplify[std]{a / b + {s1} * (c / d) = (ad + {s1} * bc) / bd}\]

and for this exercise we have $\simplify{a={a}x+{b1}}$, $\simplify{c={abs(c)}x+{abs(b2)}}$, $\simplify{b=x+{b}}$, $\simplify{d=x+{d}}$.

Hence we have:
\[\begin{eqnarray*}\simplify{({a}x+{b1}) / ({a1}*x + {b}) + ({c}x+{b2}) / ({a2}*x + {d})} &=& \simplify{(({a}x+{b1}) * ({a2}*x + {d}) + ({c}x+{b2}) * ({a1}*x + {b})) / (({a1}*x + {b}) * ({a2}*x + {d}))}\\ &=&\simplify[std]{ (({a*a2} * x^2 + {b1*a2+ a*d}x+{b1*d})+({a1*c}x^2+{b*c+a1*b2}x+{b*b2})) / (({a1}*x + {b}) * ({a2}*x + {d}))}\\&=&\simplify[std]{ ({a*a2 + c*a1} * x^2 + {a * d +a1*b2+b1*a2+ c * b}x+{b1*d+b*b2}) / (({a1}*x + {b}) * ({a2}*x + {d}))}\end{eqnarray*}\]

Question 6

Add the following two fractions together and express as a single fraction over a common denominator.

Express \[\simplify{({a}x+{b1}) / ({a1}x + {b}) + ({c}x+{b2}) / ({a2}x + {d})}\] as a single fraction.

Input the fraction here: $\displaystyle{}$ Expected answer: .

Make sure that you simplify the numerator and that you input it as a quadratic in $x$.

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Advice

The formula for taking away fractions is :

\[\simplify[std]{a / b + {s1} * (c / d) = (ad + {s1} * bc) / bd}\]

and for this exercise we have $\simplify{a={a}x+{b1}}$, $\simplify{c={abs(c)}x+{abs(b2)}}$, $\simplify{b={a1}x+{b}}$, $\simplify{d={a2}x+{d}}$.

Hence we have:
\[\begin{eqnarray*}\simplify{({a}x+{b1}) / ({a1}*x + {b}) + ({c}x+{b2}) / ({a2}*x + {d})} &=& \simplify{(({a}x+{b1}) * ({a2}*x + {d}) + ({c}x+{b2}) * ({a1}*x + {b})) / (({a1}*x + {b}) * ({a2}*x + {d}))}\\ &=&\simplify[std]{ (({a*a2} * x^2 + {b1*a2+ a*d}x+{b1*d})+({a1*c}x^2+{b*c+a1*b2}x+{b*b2})) / (({a1}*x + {b}) * ({a2}*x + {d}))}\\&=&\simplify[std]{ ({a*a2 + c*a1} * x^2 + {a * d +a1*b2+b1*a2+ c * b}x+{b1*d+b*b2}) / (({a1}*x + {b}) * ({a2}*x + {d}))}\end{eqnarray*}\]

Question 7

In the first part add the two algebraic fractions together and express as a single fraction over a common denominator.

In the second part add the three fractions together and express as a single fraction.

a)

Express \[\simplify{{a} / ({a1}x + {b}) + ({c} / ({a2}x + {d}))}\] as a single fraction.

Input the fraction here: $\displaystyle{}$ Expected answer: .

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b)

Now add the the following three fractions together to form a single fraction.

$\displaystyle \simplify{{a} / ({a1}x + {b}) + ({c} / ({a2}x + {d}))+{c1}/({a3}x+{d1})}=\;$ $\displaystyle{}$ Expected answer:

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Advice

a)

The formula for taking away fractions is :

\[\simplify[std]{a / b + {s1} * (c / d) = (ad + {s1} * bc) / bd}\]

and for this exercise we have $\simplify{b={a1}*x+{b}}$, $\simplify{d={a2}*x+{d}}$.
Hence we have:
\[\simplify[std]{{a} / ({a1}*x + {b}) + ({c} / ({a2}*x + {d})) = ({a} * ({a2}*x + {d}) + {c} * ({a1}*x + {b})) / (({a1}*x + {b}) * ({a2}*x + {d})) = ({a*a2 + c*a1} * x + {a * d + c * b}) / (({a1}*x + {b}) * ({a2}*x + {d}))}\]

b)

Note that the first two fractions are the same as in part a). Hence we immediately have:

\[\begin{eqnarray*} \simplify{{a} / ({a1}x + {b}) + ({c} / ({a2}x + {d}))+{c1}/({a3}x+{d1})}&=&\simplify[std]{({cofx} * x + {con})/ (({a1}*x + {b}) * ({a2}*x + {d}))+{c1}/({a3}x+{d1})}\\&=&\simplify[std]{((({cofx}*x+{con})({a3}x+{d1})+{c1}*({a1}*x + {b}) * ({a2}*x + {d})))/(({a1}*x + {b}) * ({a2}*x + {d})*({a3}x+{d1}))}\\&=&\simplify[std]{(({cofx*a3}x^2+{con*a3+d1*cofx}x+{con*d1})+({c1*a1*a2}x^2+{c1*(a1*d+b*a2)}x+{c1*b*d}))/(({a1}*x + {b}) * ({a2}*x + {d})*({a3}x+{d1}))}\\&=&\simplify[std]{({ans1}x^2+{ans2}x+{ans3})/(({a1}*x + {b}) * ({a2}*x + {d})*({a3}x+{d1}))}\end{eqnarray*}\]

Question 8

In the first part add the two algebraic fractions together and express as a single fraction over a common denominator.

In the second part add the three fractions together and express as a single fraction.

a)

Express \[\simplify{{a} / ({a1}x + {b}) + ({c} / ({a2}x + {d}))}\] as a single fraction.

Input the fraction here: $\displaystyle{}$ Expected answer: .

Make sure that you simplify the numerator to an expression of the form $ax+b$ for suitable integers $a$ and $b$.

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b)

Now add the the following three fractions together to form a single fraction.

$\displaystyle \simplify{{a} / ({a1}x + {b}) + ({c} / ({a2}x + {d}))+{c1}/({a3}x+{d1})}=\;$ $\displaystyle{}$ Expected answer:

Make sure that you simplify the numerator.

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Advice

a)

The formula for taking away fractions is :

\[\simplify[std]{a / b + {s1} * (c / d) = (ad + {s1} * bc) / bd}\]

and for this exercise we have $\simplify{b={a1}*x+{b}}$, $\simplify{d={a2}*x+{d}}$.
Hence we have:
\[\simplify[std]{{a} / ({a1}*x + {b}) + ({c} / ({a2}*x + {d})) = ({a} * ({a2}*x + {d}) + {c} * ({a1}*x + {b})) / (({a1}*x + {b}) * ({a2}*x + {d})) = ({a*a2 + c*a1} * x + {a * d + c * b}) / (({a1}*x + {b}) * ({a2}*x + {d}))}\]

b)

Note that the first two fractions are the same as in part a). Hence we immediately have:

\[\begin{eqnarray*} \simplify{{a} / ({a1}x + {b}) + ({c} / ({a2}x + {d}))+{c1}/({a3}x+{d1})}&=&\simplify[std]{({cofx} * x + {con})/ (({a1}*x + {b}) * ({a2}*x + {d}))+{c1}/({a3}x+{d1})}\\&=&\simplify[std]{((({cofx}*x+{con})({a3}x+{d1})+{c1}*({a1}*x + {b}) * ({a2}*x + {d})))/(({a1}*x + {b}) * ({a2}*x + {d})*({a3}x+{d1}))}\\&=&\simplify[std]{(({cofx*a3}x^2+{con*a3+d1*cofx}x+{con*d1})+({c1*a1*a2}x^2+{c1*(a1*d+b*a2)}x+{c1*b*d}))/(({a1}*x + {b}) * ({a2}*x + {d})*({a3}x+{d1}))}\\&=&\simplify[std]{({ans1}x^2+{ans2}x+{ans3})/(({a1}*x + {b}) * ({a2}*x + {d})*({a3}x+{d1}))}\end{eqnarray*}\]

Question 9

In the first part add the two algebraic fractions together and express as a single fraction over a common denominator.

In the second part add the three fractions together and express as a single fraction.

a)

Express \[\simplify{{a} / ({a1}x + {b}) + ({c} / ({a2}x + {d}))}\] as a single fraction.

Input the fraction here: $\displaystyle{}$ Expected answer: .

Make sure that you simplify the numerator.

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b)

Now add the the following three fractions together to form a single fraction.

$\displaystyle \simplify{{a} / ({a1}x + {b}) + ({c} / ({a2}x + {d}))+{c1}/({a3}x+{d1})}=\;$ $\displaystyle{}$ Expected answer:

Make sure that you simplify the numerator.

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Advice

a)

The formula for taking away fractions is :

\[\simplify[std]{a / b + {s1} * (c / d) = (ad + {s1} * bc) / bd}\]

and for this exercise we have $\simplify{b={a1}*x+{b}}$, $\simplify{d={a2}*x+{d}}$.
Hence we have:
\[\simplify[std]{{a} / ({a1}*x + {b}) + ({c} / ({a2}*x + {d})) = ({a} * ({a2}*x + {d}) + {c} * ({a1}*x + {b})) / (({a1}*x + {b}) * ({a2}*x + {d})) = ({a*a2 + c*a1} * x + {a * d + c * b}) / (({a1}*x + {b}) * ({a2}*x + {d}))}\]

b)

Note that the first two fractions are the same as in part a). Hence we immediately have:

\[\begin{eqnarray*} \simplify{{a} / ({a1}x + {b}) + ({c} / ({a2}x + {d}))+{c1}/({a3}x+{d1})}&=&\simplify[std]{({cofx} * x + {con})/ (({a1}*x + {b}) * ({a2}*x + {d}))+{c1}/({a3}x+{d1})}\\&=&\simplify[std]{((({cofx}*x+{con})({a3}x+{d1})+{c1}*({a1}*x + {b}) * ({a2}*x + {d})))/(({a1}*x + {b}) * ({a2}*x + {d})*({a3}x+{d1}))}\\&=&\simplify[std]{(({cofx*a3}x^2+{con*a3+d1*cofx}x+{con*d1})+({c1*a1*a2}x^2+{c1*(a1*d+b*a2)}x+{c1*b*d}))/(({a1}*x + {b}) * ({a2}*x + {d})*({a3}x+{d1}))}\\&=&\simplify[std]{({ans1}x^2+{ans2}x+{ans3})/(({a1}*x + {b}) * ({a2}*x + {d})*({a3}x+{d1}))}\end{eqnarray*}\]

Question 10

Add the following two fractions together and express as a single fraction over a common denominator.

Express \[\simplify[std]{{a} / ({a1}x + {b}) + (({c}x+{p}) / ({a2}x^2 +{q}x+ {d}))}\] as a single fraction.

Input the fraction here: $\displaystyle{}$ Expected answer: .

Do not expand out the denominator.

Make sure that you simplify the numerator.

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Advice

The formula for adding fractions is :

\[\simplify[std]{a / b + {s1} * (c / d) = (ad + {s1} * bc) / bd}\]

and for this exercise we have $\simplify{b={a1}x+{b}}$, $\simplify{d={a2}x^2+{q}x+{d}}$.
Hence we have:
\[\begin{eqnarray*}\simplify[std]{{a} / ({a1}*x + {b}) + (({c}x+{p}) / ({a2}*x^2+{q}x + {d}))} &=& \simplify[std]{({a} * ({a2}*x^2 +{q}x+ {d}) + ({c}x+{p}) * ({a1}*x + {b})) / (({a1}*x + {b}) * ({a2}*x^2 +{q}x+ {d}))}\\& =& \simplify[std,!collectNumbers]{({a*a2} * x^2 + {a *q}x+{a*d}+{a1*c}x^2+{p*a1+b*c}x+{b*p}) / (({a1}*x + {b}) * ({a2}*x^2 +{q}x+ {d}))}\\&=&\simplify[std,!noLeadingMinus]{({co1}x^2+{co2}x+{co3})/(({a1}*x + {b}) * ({a2}*x^2 +{q}x+ {d}))}\end{eqnarray*}\]

Question 11

Add the following two fractions together and express as a single fraction.

Express \[\simplify{{a} / ({a1}x + {b}) + ({c} / ({a2}x + {b})^2)}\] as a single fraction.

Input the fraction here: $\displaystyle{}$ Expected answer: .

Make sure that you simplify the numerator.

Click on Show steps if you require help. You will lose one mark if you do so.

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Advice

The formula for adding fractions in this case is :

\[\simplify[std]{a / b + {s1} * (c / b^2) = (ab+ {s1} * c) / b^2}\]

and for this exercise we have $\simplify{b={a1}x+{b}}$.
Hence we have:
\[\simplify[std]{{a} / ({a1}*x + {b}) + ({c} / ({a1}*x + {b})^2) = ({a} * ({a1}*x + {b}) + {c} ) / (({a1}*x + {b})^2) = ({a*a1} * x + {a * b + c}) / (({a1}*x + {b})^2 )}\]

Question 12

Add the following two fractions together and express as a single fraction.

Express \[\simplify[std]{{a} / ({a1}x + {b}) + ({c}x+{d}) /( ({a2}x + {b})^2)}\] as a single fraction.

Input the fraction here: $\displaystyle{}$ Expected answer: .

Make sure that you simplify the numerator.

Click on Show steps if you require help. You will lose one mark if you do so.

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Advice

Using the information given by Show steps we have:

\[\simplify[std]{ {a} / ({a1}x + {b}) + ({c}x+{d}) /( ({a2}x + {b})^2) = ({a} * ({a1}*x + {b}) + {c}*x+{d} ) / (({a1}*x + {b})^2) = ({a*a1+c} * x + {a * b + d}) / (({a1}*x + {b})^2 )}\]

Question 13

Add the following two fractions together and express as a single fraction over a common denominator.

Express \[\simplify{{a} /((x+{p}) ({a1}x + {b})) + ({c} /( (x+{p})({a2}x + {d})))}\] as a single fraction.

Input the fraction here: $\displaystyle{}$ Expected answer: .

Make sure that you simplify the numerator.

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Advice

Show steps tells us that a good choice for the denominator of the algebraic fraction we are looking for is $\simplify{(x+{p})({a1}x+{b})({a2}x+{d})}$.

Hence we have:
\[\begin{eqnarray*} \simplify{{a} /((x+{p}) ({a1}x + {b})) + ({c} /( (x+{p})({a2}x + {d})))} &=& \simplify[std]{({a} * ({a2}*x + {d}) + {c} * ({a1}*x + {b})) / ((x+{p})({a1}*x + {b}) * ({a2}*x + {d})) }\\&=& \simplify{({a*a2 + c*a1} * x + {a * d + c * b}) / ((x+{p})({a1}*x + {b}) * ({a2}*x + {d}))}\end{eqnarray*}\]

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mathcentre: Combining algebraic fractions

Question 1

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Question 2

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Question 3

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Question 4

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Question 5

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Question 6

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Question 7

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b)

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Question 8

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b)

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Question 9

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b)

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Question 10

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Question 11

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Question 12

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Question 13

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