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Factor Theorem and Remainder Theorem

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

The remainder theorem says f(c) is the remainder when f(x) is divided by x - c. The factor theorem says x - c is a factor exactly when f(c) = 0.

What you'll learn

Why it matters: Verifying that a known value is a root, designing polynomials with specific zeros, and shrinking a polynomial after one factor is found all rely on the factor theorem. P(a) = 0 if and only if (x − a) is a factor — that one-line rule connects algebra and graphs.

Worked example

Problem. For f(x) = x^2 - 5x + 6, is x - 2 a factor?

  1. Use c = 2.
  2. f(2) = 4 - 10 + 6 = 0.
  3. Because f(2) = 0, x - 2 is a factor.

Answer: yes

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

Practice problems

1. If f(3) = 0, which expression is a factor?

Choices: x - 3 · x + 3 · 3x · x/3

Show solution
  1. A zero at 3 gives factor x - 3.

Answer: x - 3

2. For f(x) = x^2 - 9, find f(3).

Show solution
  1. 9 - 9 = 0.

Answer: 0

3. For f(x) = x^2 + x - 6, is x - 2 a factor?

Choices: Yes · No

Show solution
  1. f(2) = 4 + 2 - 6 = 0.

Answer: Yes

4. If x + 4 is a factor, what zero does it give?

Show solution
  1. x + 4 = x - (-4).

Answer: -4

5. If f(2) = 0, which factor must f(x) have?

Choices: x - 2 · x + 2 · 2x · x - 0

Show solution
  1. The factor theorem says f(c) = 0 means x - c is a factor.
  2. Here c = 2.
  3. So x - 2 must be a factor.

Answer: x - 2

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