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Polynomial Division

A free College Algebra lesson from the “Polynomial Arithmetic and Theorems” unit, with a worked example and practice problems including step-by-step solutions.

Polynomial division rewrites one polynomial divided by another. Dividing by a monomial means dividing each term. A zero remainder indicates a factor.

What you'll learn

Why it matters: Polynomial division helps reduce a model after one factor is known, organize remainders, and connect roots to factors. The setup matters because a zero remainder is structural evidence that the divisor fits exactly.

Worked example

Problem. Divide (12x^3 + 6x^2) by 3x^2.

  1. Divide each term by 3x^2.
  2. 12x^3/3x^2 = 4x.
  3. 6x^2/3x^2 = 2.

Answer: 4x + 2

Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (typed response and multiple choice). Counts include repeated prompts and variants.

Practice problems

1. Divide 15x^4 by 5x.

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  1. 15/5 = 3 and x^4/x = x^3.

Answer: 3x^3

2. Divide (10x^3 - 5x^2) by 5x^2.

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  1. Divide each term.

Answer: 2x - 1

3. If division by x - 4 has remainder 0, then x - 4 is a...

Choices: Factor · Vertex · Coefficient · Domain

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  1. Zero remainder means exact division.

Answer: Factor

4. Divide (x^2 + 5x + 6) by (x + 2).

Choices: x + 3 · x + 2 · x - 3 · x^2 + 3

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  1. Factor the numerator as (x + 2)(x + 3).

Answer: x + 3

5. For division by x - 4, what number is used in synthetic division?

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  1. Set x - 4 = 0.

Answer: 4

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