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Rational Expressions

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

A rational expression is a fraction with polynomials. Factor numerator and denominator first, cancel only common factors, and keep excluded values from the original denominator.

What you'll learn

Why it matters: Mixing ratios, work-rate problems, and electrical-resistance sums combine rational expressions. Factoring before adding is what reveals the common denominator, and noting excluded values is what keeps the answer mathematically honest.

Worked example

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

  1. Factor x^2 - 9 as (x - 3)(x + 3).
  2. Cancel the common factor x - 3.
  3. The original denominator means x cannot be 3.

Answer: x + 3, x != 3

Practice problems

1. Simplify (x^2 - 16)/(x - 4).

Choices: x + 4, x != 4 · x - 4 · x + 4, x != -4 · x^2 + 4

Show solution
  1. Warm-up: First identify exactly what the question is asking: Simplify (x^2 - 16)/(x - 4).
  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 + 4, x != 4

2. For 1/(x + 7), what value is excluded?

Show solution
  1. Warm-up: First identify exactly what the question is asking: For 1/(x + 7), 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. The denominator cannot be zero.
  4. Check the result by substituting or estimating: the response should match -7 and make sense in the original problem.

Answer: -7

3. Simplify (x^2 + 5x)/(x).

Choices: x + 5, x != 0 · x^2 + 5 · 5x · x + 5, x != -5

Show solution
  1. Core Practice: First identify exactly what the question is asking: Simplify (x^2 + 5x)/(x).
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. Factor x(x + 5), then cancel x.
  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 != 0

4. Simplify (x^2 - x - 6)/(x - 3).

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

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

Answer: x + 2, x != 3

5. Which expression has excluded values 2 and -5?

Choices: 1/((x - 2)(x + 5)) · 1/((x + 2)(x - 5)) · (x - 2)/(x + 5) · (x + 5)/(x - 2)

Show solution
  1. Challenge: First identify exactly what the question is asking: Which expression has excluded values 2 and -5?
  2. For signed numbers, track both distance from zero and direction so the sign of the answer makes sense.
  3. Set each denominator factor equal to zero.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: 1/((x - 2)(x + 5))

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