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All courses Algebra II · A-APR.6 13 of 55
Rewrite rational expressions using inspection, polynomial division, or technology.

Divide polynomials using long division

Problem
Use the polynomial long-division setup to divide \(x^3~+~2x^2~-~5x~+~6\) by \(x~+~3\). State the quotient and remainder.
Unsolved polynomial long-division scaffold with divisor x+3 and dividend x^3+2x^2-5x+6. The quotient line and all subtract-and-bring-down work are blank; align like powers. Quotient and remainder are withheld. 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

Divide the leading terms

\[x^3/x=x^2\]

The dividend x³+2x²-5x+6 is already in descending powers. Dividing its leading term by the divisor's leading term gives the first quotient term x².

02

Multiply and subtract the first row

\[(x^3+2x^2-5x+6)-(x^3+3x^2)=-x^2-5x+6\]

Multiply x+3 by x² to get x³+3x². Subtracting that product cancels x³ and leaves -x²-5x+6.

03

Continue through the remaining powers

\[-x^2/x=-x;~(-2x)/x=-2\]

The next quotient term is -x. Subtracting (-x)(x+3)=-x²-3x leaves -2x+6. Then -2 is the last quotient term, and subtracting -2(x+3)=-2x-6 leaves 12.

04

State and verify quotient and remainder

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

The division identity is x³+2x²-5x+6=(x+3)(x²-x-2)+12. Expanding the product gives x³+2x²-5x-6, and adding 12 restores the dividend, so the quotient and remainder are correct, choice A.

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Another way

  1. Use synthetic division with -3 on coefficients 1, 2, -5, 6; the bottom row 1, -1, -2, 12 gives the same quotient and remainder.

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Common mistake

When subtracting a product row, distribute the subtraction to every term. In the last row, (-2x+6)-(-2x-6)=12, not 0.

Completed polynomial long division of x^3+2x^2-5x+6 by x+3: quotient x^2-x-2 and remainder 12. Equivalently, x^3+2x^2-5x+6=(x+3)(x^2-x-2)+12.
Aligned work distinguishes quotient from remainder.

Solution walkthrough video