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Polynomial Division

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

Polynomial division rewrites a rational-looking expression as quotient plus remainder over divisor. 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. Divide x^2 + 6x + 9 by (x + 3). Enter the quotient.

  1. Notice x^2 + 6x + 9 is a perfect square: (x + 3)^2.
  2. So (x + 3)(x + 3) = x^2 + 6x + 9, dividing evenly.
  3. The quotient is x + 3 with remainder 0.

Answer: x + 3

Practice problems

1. Divide x^2 + 6x + 9 by (x + 3). Enter the quotient.

Show solution
  1. Notice x^2 + 6x + 9 is a perfect square: (x + 3)^2.
  2. So (x + 3)(x + 3) = x^2 + 6x + 9, dividing evenly.
  3. The quotient is x + 3 with remainder 0.

Answer: x + 3

2. Divide x^2 + 8x + 19 by (x + 4) using long division. Enter the remainder.

Show solution
  1. x^2 + 8x + 16 equals (x + 4)^2 and divides evenly.
  2. That leaves the extra constant + 3, whose degree is below the divisor.
  3. So the quotient is x + 4 and the remainder is 3.

Answer: 3

3. Divide x^2 - 25 by (x - 5). Enter the quotient.

Show solution
  1. x^2 - 25 is a difference of squares: (x - 5)(x + 5).
  2. Dividing by (x - 5) cancels that factor.
  3. The quotient is x + 5 with remainder 0.

Answer: x + 5

4. By the Remainder Theorem, find the remainder when p(x) = x^2 + 1 is divided by (x - 2).

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 + 1 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 Theorem says the remainder equals p(2).
  4. p(2) = (2)^2 + 1 = 4 + 1.
  5. So the remainder is 5.
  6. Check the result by substituting or estimating: the response should match 5 and make sense in the original problem.

Answer: 5

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

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 + 2 is divided by (x + 3).
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Dividing by (x + 3) means evaluating p at x = -3.
  4. p(-3) = (-3)^2 + 2 = 9 + 2, since (-3)^2 = 9.
  5. 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

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