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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

Practice problems

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

Choices: x - 3 · x + 3 · 3x · x/3

Show solution
  1. Warm-up: First identify exactly what the question is asking: If f(3) = 0, which expression is a factor?
  2. Use the structure of the expression to choose a factoring pattern, then check that the factors multiply back to the original expression.
  3. A zero at 3 gives factor 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

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

Show solution
  1. Warm-up: First identify exactly what the question is asking: For f(x) = x^2 - 9, find f(3).
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. 9 - 9 = 0.
  4. Check the result by substituting or estimating: the response should match 0 and make sense in the original problem.

Answer: 0

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

Choices: Yes · No

Show solution
  1. Core Practice: First identify exactly what the question is asking: For f(x) = x^2 + x - 6, is x - 2 a factor?
  2. Use the structure of the expression to choose a factoring pattern, then check that the factors multiply back to the original expression.
  3. f(2) = 4 + 2 - 6 = 0.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: Yes

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

Show solution
  1. Challenge: First identify exactly what the question is asking: If x + 4 is a factor, what zero does it give?
  2. Use the structure of the expression to choose a factoring pattern, then check that the factors multiply back to the original expression.
  3. x + 4 = x - (-4).
  4. Check the result by substituting or estimating: the response should match -4 and make sense in the original problem.

Answer: -4

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

Choices: x - 2 · x + 2 · 2x · x - 0

Show solution
  1. Mixed Review: First identify exactly what the question is asking: If f(2) = 0, which factor must f(x) have?
  2. Use the structure of the expression to choose a factoring pattern, then check that the factors multiply back to the original expression.
  3. The factor theorem says f(c) = 0 means x - c is a factor.
  4. Here c = 2.
  5. So x - 2 must be a factor.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: x - 2

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