Sec 3 A-Math · Polynomials, cubic equations and partial fractions

Dividing polynomials questions with answers

Practice questions on dividing polynomials, from the basics up to exam level, each with a full worked solution. Try them on paper first, then open the solution to check your working.

Worked example

Find the quotient when $x^3 - 6x^2 + 10x - 4$ is divided by $x - 2$ .

Step 1 of 2

Long division: divide the leading terms, multiply the divisor by that term, subtract, bring down the next term; repeat

\def\arraystretch{1.6}\begin{array}{rrrrr} & & \;x^2\; & \;-\,4x\; & \;+\,2\; \\ x - 2\,\Big) & \overline{\;x^3\;} & \overline{\;-\,6x^2\;} & \overline{\;+\,10x\;} & \overline{\;-\,4\;} \\ & \underline{\;x^3\;} & \underline{\;-\,2x^2\;} & & \\ & & \;-4x^2\; & \;+\,10x\; & \\ & & \underline{\;-4x^2\;} & \underline{\;+\,8x\;} & \\ & & & \;2x\; & \;-\,4\; \\ & & & \underline{\;2x\;} & \underline{\;-\,4\;} \\ & & & & \;0\; \end{array}

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Questions

Question 1Exact division: the quotient
Find the quotient when $x^3 - 2x^2 - 6x + 4$ is divided by $x + 2$ .
Show the worked solution
1. Step 1: Long division: divide the leading terms, multiply the divisor by that term, subtract, bring down the next term; repeat $\def\arraystretch{1.6}\begin{array}{rrrrr} & & \;x^2\; & \;-\,4x\; & \;+\,2\; \\ x + 2\,\Big) & \overline{\;x^3\;} & \overline{\;-\,2x^2\;} & \overline{\;-\,6x\;} & \overline{\;+\,4\;} \\ & \underline{\;x^3\;} & \underline{\;+\,2x^2\;} & & \\ & & \;-4x^2\; & \;-\,6x\; & \\ & & \underline{\;-4x^2\;} & \underline{\;-\,8x\;} & \\ & & & \;2x\; & \;+\,4\; \\ & & & \underline{\;2x\;} & \underline{\;+\,4\;} \\ & & & & \;0\; \end{array}$
2. Step 2: The quotient is the top line (the remainder at the bottom is 0) $\text{Quotient} = x^2 - 4x + 2$
Question 2Exact division: the quotient
Find the quotient when $x^3 + x^2 - 5x - 2$ is divided by $x - 2$ .
Show the worked solution
1. Step 1: Long division: divide the leading terms, multiply the divisor by that term, subtract, bring down the next term; repeat $\def\arraystretch{1.6}\begin{array}{rrrrr} & & \;x^2\; & \;+\,3x\; & \;+\,1\; \\ x - 2\,\Big) & \overline{\;x^3\;} & \overline{\;+\,x^2\;} & \overline{\;-\,5x\;} & \overline{\;-\,2\;} \\ & \underline{\;x^3\;} & \underline{\;-\,2x^2\;} & & \\ & & \;3x^2\; & \;-\,5x\; & \\ & & \underline{\;3x^2\;} & \underline{\;-\,6x\;} & \\ & & & \;x\; & \;-\,2\; \\ & & & \underline{\;x\;} & \underline{\;-\,2\;} \\ & & & & \;0\; \end{array}$
2. Step 2: The quotient is the top line (the remainder at the bottom is 0) $\text{Quotient} = x^2 + 3x + 1$
Question 3The remainder
Find the remainder when $x^3 - 4x^2 + 7x - 14$ is divided by $x - 3$ .
Show the worked solution
1. Step 1: Long division: divide the leading terms, multiply the divisor by that term, subtract, bring down the next term; repeat $\def\arraystretch{1.6}\begin{array}{rrrrr} & & \;x^2\; & \;-\,x\; & \;+\,4\; \\ x - 3\,\Big) & \overline{\;x^3\;} & \overline{\;-\,4x^2\;} & \overline{\;+\,7x\;} & \overline{\;-\,14\;} \\ & \underline{\;x^3\;} & \underline{\;-\,3x^2\;} & & \\ & & \;-x^2\; & \;+\,7x\; & \\ & & \underline{\;-x^2\;} & \underline{\;+\,3x\;} & \\ & & & \;4x\; & \;-\,14\; \\ & & & \underline{\;4x\;} & \underline{\;-\,12\;} \\ & & & & \;-2\; \end{array}$
2. Step 2: The remainder is what is left at the bottom $\text{Remainder} = -2$
Question 4The remainder
Find the remainder when $x^3 + x^2 - 5x + 10$ is divided by $x - 1$ .
Show the worked solution
1. Step 1: Long division: divide the leading terms, multiply the divisor by that term, subtract, bring down the next term; repeat $\def\arraystretch{1.6}\begin{array}{rrrrr} & & \;x^2\; & \;+\,2x\; & \;-\,3\; \\ x - 1\,\Big) & \overline{\;x^3\;} & \overline{\;+\,x^2\;} & \overline{\;-\,5x\;} & \overline{\;+\,10\;} \\ & \underline{\;x^3\;} & \underline{\;-\,x^2\;} & & \\ & & \;2x^2\; & \;-\,5x\; & \\ & & \underline{\;2x^2\;} & \underline{\;-\,2x\;} & \\ & & & \;-3x\; & \;+\,10\; \\ & & & \underline{\;-3x\;} & \underline{\;+\,3\;} \\ & & & & \;7\; \end{array}$
2. Step 2: The remainder is what is left at the bottom $\text{Remainder} = 7$
Question 5Identities: compare coefficients
Given that $3x^{2} + 7x + 9 \equiv A(x + 2)^2 + B(x + 2) + C$ for all values of $x$ , find the values of $A$ , $B$ and $C$ .
Show the worked solution
1. Step 1: Compare the coefficients of x² (only the squared bracket has an x² term) $A = 3$
2. Step 2: Put x = -2: both brackets are 0 $3(-2)^{2} + 7(-2) + 9 = C$ $C = 7$
3. Step 3: Compare the coefficients of x: expand the squared bracket in your head $7 = 4A + B$ $B = -5$
Question 6Identities: compare coefficients
Given that $2x^{2} - 8x + 5 \equiv A(x - 3)^2 + B(x - 3) + C$ for all values of $x$ , find the values of $A$ , $B$ and $C$ .
Show the worked solution
1. Step 1: Compare the coefficients of x² (only the squared bracket has an x² term) $A = 2$
2. Step 2: Put x = 3: both brackets are 0 $2(3)^{2} - 8(3) + 5 = C$ $C = -1$
3. Step 3: Compare the coefficients of x: expand the squared bracket in your head $-8 = -6A + B$ $B = 4$

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Worked example

Find the quotient when

x^3 - 6x^2 + 10x - 4

is divided by

x - 2

Step 1 of 2

Long division: divide the leading terms, multiply the divisor by that term, subtract, bring down the next term; repeat

\def\arraystretch{1.6}\begin{array}{rrrrr} & & \;x^2\; & \;-\,4x\; & \;+\,2\; \\ x - 2\,\Big) & \overline{\;x^3\;} & \overline{\;-\,6x^2\;} & \overline{\;+\,10x\;} & \overline{\;-\,4\;} \\ & \underline{\;x^3\;} & \underline{\;-\,2x^2\;} & & \\ & & \;-4x^2\; & \;+\,10x\; & \\ & & \underline{\;-4x^2\;} & \underline{\;+\,8x\;} & \\ & & & \;2x\; & \;-\,4\; \\ & & & \underline{\;2x\;} & \underline{\;-\,4\;} \\ & & & & \;0\; \end{array}

Questions

Question 1Exact division: the quotient

Find the quotient when $x^3 - 2x^2 - 6x + 4$ is divided by $x + 2$ .

Show the worked solution

Step 1: Long division: divide the leading terms, multiply the divisor by that term, subtract, bring down the next term; repeat $\def\arraystretch{1.6}\begin{array}{rrrrr} & & \;x^2\; & \;-\,4x\; & \;+\,2\; \\ x + 2\,\Big) & \overline{\;x^3\;} & \overline{\;-\,2x^2\;} & \overline{\;-\,6x\;} & \overline{\;+\,4\;} \\ & \underline{\;x^3\;} & \underline{\;+\,2x^2\;} & & \\ & & \;-4x^2\; & \;-\,6x\; & \\ & & \underline{\;-4x^2\;} & \underline{\;-\,8x\;} & \\ & & & \;2x\; & \;+\,4\; \\ & & & \underline{\;2x\;} & \underline{\;+\,4\;} \\ & & & & \;0\; \end{array}$
Step 2: The quotient is the top line (the remainder at the bottom is 0) $\text{Quotient} = x^2 - 4x + 2$

Question 2Exact division: the quotient

Find the quotient when $x^3 + x^2 - 5x - 2$ is divided by $x - 2$ .

Show the worked solution

Step 1: Long division: divide the leading terms, multiply the divisor by that term, subtract, bring down the next term; repeat $\def\arraystretch{1.6}\begin{array}{rrrrr} & & \;x^2\; & \;+\,3x\; & \;+\,1\; \\ x - 2\,\Big) & \overline{\;x^3\;} & \overline{\;+\,x^2\;} & \overline{\;-\,5x\;} & \overline{\;-\,2\;} \\ & \underline{\;x^3\;} & \underline{\;-\,2x^2\;} & & \\ & & \;3x^2\; & \;-\,5x\; & \\ & & \underline{\;3x^2\;} & \underline{\;-\,6x\;} & \\ & & & \;x\; & \;-\,2\; \\ & & & \underline{\;x\;} & \underline{\;-\,2\;} \\ & & & & \;0\; \end{array}$
Step 2: The quotient is the top line (the remainder at the bottom is 0) $\text{Quotient} = x^2 + 3x + 1$

Question 3The remainder

Find the remainder when $x^3 - 4x^2 + 7x - 14$ is divided by $x - 3$ .

Show the worked solution

Step 1: Long division: divide the leading terms, multiply the divisor by that term, subtract, bring down the next term; repeat $\def\arraystretch{1.6}\begin{array}{rrrrr} & & \;x^2\; & \;-\,x\; & \;+\,4\; \\ x - 3\,\Big) & \overline{\;x^3\;} & \overline{\;-\,4x^2\;} & \overline{\;+\,7x\;} & \overline{\;-\,14\;} \\ & \underline{\;x^3\;} & \underline{\;-\,3x^2\;} & & \\ & & \;-x^2\; & \;+\,7x\; & \\ & & \underline{\;-x^2\;} & \underline{\;+\,3x\;} & \\ & & & \;4x\; & \;-\,14\; \\ & & & \underline{\;4x\;} & \underline{\;-\,12\;} \\ & & & & \;-2\; \end{array}$
Step 2: The remainder is what is left at the bottom $\text{Remainder} = -2$

Question 4The remainder

Find the remainder when $x^3 + x^2 - 5x + 10$ is divided by $x - 1$ .

Show the worked solution

Step 1: Long division: divide the leading terms, multiply the divisor by that term, subtract, bring down the next term; repeat $\def\arraystretch{1.6}\begin{array}{rrrrr} & & \;x^2\; & \;+\,2x\; & \;-\,3\; \\ x - 1\,\Big) & \overline{\;x^3\;} & \overline{\;+\,x^2\;} & \overline{\;-\,5x\;} & \overline{\;+\,10\;} \\ & \underline{\;x^3\;} & \underline{\;-\,x^2\;} & & \\ & & \;2x^2\; & \;-\,5x\; & \\ & & \underline{\;2x^2\;} & \underline{\;-\,2x\;} & \\ & & & \;-3x\; & \;+\,10\; \\ & & & \underline{\;-3x\;} & \underline{\;+\,3\;} \\ & & & & \;7\; \end{array}$
Step 2: The remainder is what is left at the bottom $\text{Remainder} = 7$

Question 5Identities: compare coefficients

Given that $3x^{2} + 7x + 9 \equiv A(x + 2)^2 + B(x + 2) + C$ for all values of $x$ , find the values of $A$ , $B$ and $C$ .

Show the worked solution

Step 1: Compare the coefficients of x² (only the squared bracket has an x² term) $A = 3$
Step 2: Put x = -2: both brackets are 0 $3(-2)^{2} + 7(-2) + 9 = C$ $C = 7$
Step 3: Compare the coefficients of x: expand the squared bracket in your head $7 = 4A + B$ $B = -5$

Question 6Identities: compare coefficients

Given that $2x^{2} - 8x + 5 \equiv A(x - 3)^2 + B(x - 3) + C$ for all values of $x$ , find the values of $A$ , $B$ and $C$ .

Show the worked solution

Step 1: Compare the coefficients of x² (only the squared bracket has an x² term) $A = 2$
Step 2: Put x = 3: both brackets are 0 $2(3)^{2} - 8(3) + 5 = C$ $C = -1$
Step 3: Compare the coefficients of x: expand the squared bracket in your head $-8 = -6A + B$ $B = 4$