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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 - 9, find p(3).

  1. Worked Example: First identify exactly what the question is asking: For p(x) = x^2 - 9, find p(3).
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Substitute x = 3: (3)^2 - 9.
  4. (3)^2 = 9, so 9 - 9 = 0.
  5. p(3) = 0, so by the Factor Theorem x - 3 is a factor.
  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 - 9, find p(3).

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

Answer: 0

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

Show solution
  1. Warm-up: First identify exactly what the question is asking: By the Remainder Theorem, find the remainder when p(x) = x^2 + 3 is divided by x - 2.
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. The remainder equals p(2).
  4. p(2) = (2)^2 + 3 = 4 + 3 = 7.
  5. The remainder is 7 (nonzero), so x - 2 is not a factor.
  6. Check the result by substituting or estimating: the response should match 7 and make sense in the original problem.

Answer: 7

3. By the Remainder Theorem, find the remainder when p(x) = x^2 - 25 is divided by x - 6.

Show solution
  1. Core Practice: First identify exactly what the question is asking: By the Remainder Theorem, find the remainder when p(x) = x^2 - 25 is divided by x - 6.
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. The remainder equals p(6).
  4. p(6) = (6)^2 - 25 = 36 - 25.
  5. 36 - 25 = 11, so the remainder is 11.
  6. Check the result by substituting or estimating: the response should match 11 and make sense in the original problem.

Answer: 11

4. For p(x) = x^2 + 3x - 1, find p(2).

Show solution
  1. Core Practice: First identify exactly what the question is asking: For p(x) = x^2 + 3x - 1, find p(2).
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Substitute x = 2: (2)^2 + 3(2) - 1.
  4. (2)^2 = 4 and 3(2) = 6.
  5. 4 + 6 - 1 = 9.
  6. Check the result by substituting or estimating: the response should match 9 and make sense in the original problem.

Answer: 9

5. For p(x) = x^2 - 4x, find p(-2).

Show solution
  1. Core Practice: First identify exactly what the question is asking: For p(x) = x^2 - 4x, find p(-2).
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Substitute x = -2: (-2)^2 - 4(-2).
  4. (-2)^2 = +4 (a square is positive) and -4(-2) = +8.
  5. 4 + 8 = 12.
  6. Check the result by substituting or estimating: the response should match 12 and make sense in the original problem.

Answer: 12

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