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
- Simplify rational expressions
- Find excluded values
- Factor before canceling
Worked example
Problem. Simplify (x^2 - 25)/(x - 5).
- Factor x^2 - 25 as (x - 5)(x + 5).
- Cancel x - 5.
- The original denominator excludes x = 5.
Answer: x + 5, x != 5
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 - 9)/(x - 3).
Choices: x + 3, x != 3 · x - 3 · x + 3, x != -3 · x^2 + 3
Show solution
- Factor difference of squares.
Answer: x + 3, x != 3
2. For 1/(x + 12), what value is excluded?
Show solution
- x + 12 cannot be zero.
Answer: -12
3. Simplify (x^2 + 6x)/(x).
Choices: x + 6, x != 0 · x^2 + 6 · 6x · x + 6, x != -6
Show solution
- Factor x(x + 6).
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
- Factor the numerator as (x - 5)(x + 5).
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
- The denominator factors as (x - 4)(x + 4).
Answer: x = -4 and x = 4
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