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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

Practice problems

1. Divide 15x^4 by 5x.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Divide 15x^4 by 5x.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. 15/5 = 3 and x^4/x = x^3.
  4. Check the result by substituting or estimating: the response should match 3x^3 and make sense in the original problem.

Answer: 3x^3

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

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  1. Core Practice: First identify exactly what the question is asking: Divide (10x^3 - 5x^2) by 5x^2.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Divide each term.
  4. Check the result by substituting or estimating: the response should match 2x - 1 and make sense in the original problem.

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. Challenge: First identify exactly what the question is asking: If division by x - 4 has remainder 0, then x - 4 is a...
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. Zero remainder means exact division.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: Factor

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

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

Show solution
  1. Factored Division: First identify exactly what the question is asking: Divide (x^2 + 5x + 6) by (x + 2).
  2. Use the structure of the expression to choose a factoring pattern, then check that the factors multiply back to the original expression.
  3. Factor the numerator as (x + 2)(x + 3).
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: x + 3

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

Show solution
  1. Synthetic Division: First identify exactly what the question is asking: For division by x - 4, what number is used in synthetic division?
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Set x - 4 = 0.
  4. Check the result by substituting or estimating: the response should match 4 and make sense in the original problem.

Answer: 4

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