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All courses Math III · A-APR.6 6 of 55
Rewrite rational expressions using inspection, polynomial division, or technology.

Divide polynomials using long division

Problem
Divide these polynomials using long division: (x³+2x²-5x+6)/(x+3).
Unsolved prompt rational long division. Open full size
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Hint

Divide the leading term \(x^3\) by the leading term \(x\) to get the first quotient term.

In long division, divide, multiply, subtract, and then bring down the next term.

Solution walkthrough

01

Find the first quotient term

\[x^3~÷~x~=~x^2\]

The quotient starts with \(x^2\) because the leading term of the dividend divided by the leading term of the divisor is \(x^2\).

02

Multiply and subtract

\[(x+3)(x^2)~=~x^3+3x^2,~\text{so}~(x^3+2x^2-5x+6)-(x^3+3x^2)~=~-x^2-5x+6\]

After placing \(x^2\) in the quotient, subtract the matching product from the dividend.

03

Repeat the division steps

\[-x^2~÷~x~=~-x,~\text{then}~-2x~÷~x~=~-2\]

Continuing the divide-multiply-subtract pattern gives the remaining quotient terms \(-x\) and \(-2\).

04

State the quotient and remainder

\[\text{quotient}~x^2-x-2,~\text{remainder}~12\]

After the last subtraction, the leftover value is \(12\), so the result is quotient \(x^2-x-2\) with remainder \(12\).

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Another way

  1. You can check the remainder with the Remainder Theorem: \(p(-3)=12\) for divisor \(x+3\).

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Common mistake

A common mistake is to forget to subtract the entire product \((x^3+3x^2)\), which changes the sign of the \(3x^2\) term incorrectly.

Completed answer rational long division.
Aligned work distinguishes quotient from remainder.