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Rational Expressions

A free College Algebra lesson from the “Rational Expressions and Equations” unit, with a worked example and practice problems including step-by-step solutions.

Rational expressions are fractions with polynomials. Factor first, cancel common factors, and keep excluded values from the original denominator.

What you'll learn

Why it matters: Rates, work problems, ratios, and formulas with restrictions often become rational expressions. Factoring before canceling reveals what simplifies and what input values were never allowed.

Worked example

Problem. Simplify (x^2 - 25)/(x - 5).

  1. Factor x^2 - 25 as (x - 5)(x + 5).
  2. Cancel x - 5.
  3. The original denominator excludes x = 5.

Answer: x + 5, x != 5

Practice problems

1. Simplify (x^2 - 9)/(x - 3).

Choices: x + 3, x != 3 · x - 3 · x + 3, x != -3 · x^2 + 3

Show solution
  1. Warm-up: First identify exactly what the question is asking: Simplify (x^2 - 9)/(x - 3).
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. Factor difference of squares.
  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, x != 3

2. For 1/(x + 12), what value is excluded?

Show solution
  1. Core Practice: First identify exactly what the question is asking: For 1/(x + 12), what value is excluded?
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. x + 12 cannot be zero.
  4. Check the result by substituting or estimating: the response should match -12 and make sense in the original problem.

Answer: -12

3. Simplify (x^2 + 6x)/(x).

Choices: x + 6, x != 0 · x^2 + 6 · 6x · x + 6, x != -6

Show solution
  1. Challenge: First identify exactly what the question is asking: Simplify (x^2 + 6x)/(x).
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. Factor x(x + 6).
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: x + 6, x != 0

4. Simplify (x^2 - 25)/(x - 5).

Choices: x + 5, x != 5 · x - 5 · x + 5, x != -5 · x^2 + 5

Show solution
  1. Simplifying: First identify exactly what the question is asking: Simplify (x^2 - 25)/(x - 5).
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. Factor the numerator as (x - 5)(x + 5).
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: x + 5, x != 5

5. For (x + 2)/(x^2 - 16), which values are excluded?

Choices: x = -4 and x = 4 · x = 2 only · x = -2 only · No values

Show solution
  1. Restrictions: First identify exactly what the question is asking: For (x + 2)/(x^2 - 16), which values are excluded?
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. The denominator factors as (x - 4)(x + 4).
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: x = -4 and x = 4

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