Algebra 2
Rational functions
Simplifying rational expressions
20 practice questions
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Theory + worked examples
Simplifying Rational Expressions
California Algebra 2 • Standard A-APR.6 • Rational Functions
Simplifying Rational Expressions is a topic in Rational Functions in the California Common Core State Standards. It is aligned to Standard A-APR.6, which requires students to rewrite simple rational expressions in different forms.
Simplifying factors the numerator and denominator, cancels common factors, and states the excluded values.
Theory
To simplify a rational expression:
- Factor the numerator and denominator completely.
- Cancel factors common to both.
- State the excluded values from the original denominator.
Only cancel factors, never individual terms across a \(+\) or \(-\).
Factor, cancel, then restrict.
A worked simplification.
The rule:
\[\dfrac{a\cdot c}{b\cdot c}=\dfrac{a}{b}\quad(c\neq0)\]
Keep the excluded values from the cancelled factor.
How to simplify
- Factor everything.
- Cancel common factors.
- Note the excluded values.
- Write the reduced expression with its restrictions.
Example 1 — Difference of squares
Simplify \(\dfrac{x^2-9}{x+3}\).
Solution
Factor and cancel.
| \(\dfrac{(x-3)(x+3)}{x+3}\) | \(=\) | \(x-3,\ x\neq-3\) |
Example 2 — Two trinomials
Simplify \(\dfrac{x^2+5x+6}{x^2-4}\).
Solution
Factor both.
| \(\dfrac{(x+2)(x+3)}{(x-2)(x+2)}\) | \(=\) | \(\dfrac{x+3}{x-2}\) |
Example 3 — Factor out a constant
Simplify \(\dfrac{2x^2-8}{x^2-4x+4}\).
Solution
Factor the common \(2\) and the perfect square.
| \(\dfrac{2(x-2)(x+2)}{(x-2)^2}\) | \(=\) | \(\dfrac{2(x+2)}{x-2}\) |
Example 4 — State the restriction
What restriction applies to the answer in Example 1?
Solution
The cancelled factor still restricts the domain.
| \(x\) | \(\neq\) | \(-3\) |
Common pitfalls
Cancel factors, not terms: you can't cancel the \(x\) in \(\dfrac{x+2}{x}\).
Keep the restriction from a cancelled factor.
Factor completely before cancelling.
Frequently asked questions
How do you simplify a rational expression?
Factor top and bottom, cancel common factors, and state restrictions.
Can you cancel terms?
No — only common factors, never terms across addition.
Why keep the excluded values?
The domain restriction survives even after a factor cancels.
Does \(\dfrac{x+2}{x}\) simplify?
No — \(x\) is not a factor of the numerator.
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Asymptotes (vertical, horizontal, removable)
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Operations on rational expressions
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