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Remainder and Factor Theorems

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

If p(c) = 0, then x - c is a factor. If p(c) is not zero, that value is the remainder. This lesson is part of Precalculus: Advanced Functions, so the emphasis is on interpreting behavior, choosing the right representation, and explaining the result clearly rather than memorizing isolated algebra moves.

What you'll learn

Why it matters: Polynomial models describe smooth turning behavior in design, approximation, data fitting, and STEM modeling.

Worked example

Problem. For p(x) = x^2 - 1, find p(-1).

  1. Worked Example: First identify exactly what the question is asking: For p(x) = x^2 - 1, find p(-1).
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Substitute -1 into p(x).
  4. (-1)^2 - 1 = 0.
  5. A value of 0 means x minus that input is a factor (Factor Theorem).
  6. Check the result by substituting or estimating: the response should match 0 and make sense in the original problem.

Answer: 0

Practice problems

1. For p(x) = x^2 - 1, find p(-1).

Show solution
  1. Warm-up: First identify exactly what the question is asking: For p(x) = x^2 - 1, find p(-1).
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Substitute -1 into p(x).
  4. (-1)^2 - 1 = 0.
  5. A value of 0 means x minus that input is a factor (Factor Theorem).
  6. Check the result by substituting or estimating: the response should match 0 and make sense in the original problem.

Answer: 0

2. For p(x) = x^2 - 0, find p(0).

Show solution
  1. Warm-up: First identify exactly what the question is asking: For p(x) = x^2 - 0, find p(0).
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Substitute 0 into p(x).
  4. (0)^2 - 0 = 0.
  5. A value of 0 means x minus that input is a factor (Factor Theorem).
  6. Check the result by substituting or estimating: the response should match 0 and make sense in the original problem.

Answer: 0

3. By the Remainder Theorem, find the remainder when p(x) = x^2 + 2 is divided by x - 2.

Show solution
  1. The Remainder Theorem says the remainder equals p(2).
  2. p(2) = (2)^2 + 2 = 6.
  3. The remainder 6 is not 0, so x - 2 is not a factor.

Answer: 6

4. If p(2) = 0, what does the Factor Theorem say?

Choices: x - (2) is a factor · x + 2 is never a factor · p has no zeros · the degree must be 0

Show solution
  1. Core Practice: First identify exactly what the question is asking: If p(2) = 0, what does the Factor Theorem say?
  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 links p(c) = 0 with factor x - c.
  4. Here c = 2.
  5. So x - c is 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) is a factor

5. If p(-2) = 0, what does the Factor Theorem say?

Choices: x - (-2) is a factor · x + -2 is never a factor · p has no zeros · the degree must be 0

Show solution
  1. Core Practice: First identify exactly what the question is asking: If p(-2) = 0, what does the Factor Theorem say?
  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 links p(c) = 0 with factor x - c.
  4. Here c = -2.
  5. So x - c is 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) is a factor

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