How to simplify rational expressions
Quick answer
Factor the numerator and the denominator completely, cancel any factors that appear in both, and write down the excluded values, the x values that would make the original denominator zero. For example, (x² − 9) / (x² + 5x + 6) simplifies to (x − 3) / (x + 2), with x ≠ −2 and x ≠ −3.
Step by step
- Factor the topFactor the numerator completely, using common factors, differences of squares or trinomial factoring.
- Factor the bottomDo the same for the denominator.
- Find excluded valuesSet each factor of the original denominator equal to zero. Those x values are not allowed.
- Cancel common factorsDivide out any factor that appears in both the numerator and the denominator.
- Write the answerWrite what’s left, along with the excluded values.
A rational expression is just a fraction with polynomials on top and bottom. Simplifying one works exactly like simplifying an ordinary fraction: find what the top and bottom share, and divide it out.
Example 1: A common factor
(2x + 6) / (4x + 12)
Factor: 2(x + 3) / 4(x + 3)
Cancel (x + 3) and simplify 2/4: 1/2, with x ≠ −3
Example 2: Difference of squares and a trinomial
(x² − 9) / (x² + 5x + 6)
Top: x² − 9 = (x − 3)(x + 3)
Bottom: x² + 5x + 6 = (x + 2)(x + 3)
Cancel (x + 3): (x − 3) / (x + 2), with x ≠ −2 and x ≠ −3
Example 3: Opposite factors
(5 − x) / (x − 5)
Factor out −1 from the top: −(x − 5) / (x − 5)
Answer: −1, with x ≠ 5
Watch for this pattern: (a − b) and (b − a) are opposites, so they cancel to −1, not 1.
Why the excluded values matter
The simplified expression may be defined at a point where the original wasn’t. In Example 2, (x − 3)/(x + 2) works fine at x = −3, but the original expression doesn’t, because its denominator would be zero there. Listing excluded values keeps the two expressions truly equal.
If factoring trinomials is the sticking point, practice simplifying algebraic expressions first.
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