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
- Factor rational expressions
- Simplify by canceling common factors
- Identify excluded values
Worked example
Problem. Simplify (x^2 - 9)/(x - 3).
- Factor x^2 - 9 as (x - 3)(x + 3).
- Cancel the common factor x - 3.
- The original denominator means x cannot be 3.
Answer: x + 3, x != 3
Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (multiple choice and typed response). Counts include repeated prompts and variants.
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
- Factor difference of squares.
Answer: x + 4, x != 4
2. For 1/(x + 7), what value is excluded?
Show solution
- The denominator cannot be zero.
Answer: -7
3. Simplify (x^2 + 5x)/(x).
Choices: x + 5, x != 0 · x^2 + 5 · 5x · x + 5, x != -5
Show solution
- Factor x(x + 5), then cancel x.
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
- Factor as (x - 3)(x + 2).
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
- Set each denominator factor equal to zero.
Answer: 1/((x - 2)(x + 5))
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