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Algebra 2 Rational functions

Simplifying rational expressions

20 practice questions 0 video lessons Theory + worked examples

Simplifying Rational Expressions

Texas Algebra II (TEKS) • Standard 2A.7(B) • Rational Functions

Simplifying Rational Expressions is a topic in Rational Functions in the Texas Essential Knowledge and Skills (Algebra II, §111.40). It is aligned to Standard 2A.7(B), which requires students to add, subtract, multiply, divide, and simplify rational expressions.

Simplifying factors the numerator and denominator, cancels common factors, and states the excluded values.

Texas Algebra II (TEKS) › Rational Functions › Simplifying Rational Expressions  —  Standard 2A.7(B)

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Theory

To simplify a rational expression:

  1. Factor the numerator and denominator completely.
  2. Cancel factors common to both.
  3. State the excluded values from the original denominator.
Only cancel factors, never individual terms across a \(+\) or \(-\).
Simplifying steps Simplifying steps Simplifying steps 1. factor numerator and denominator 2. cancel common factors 3. state excluded values
Factor, cancel, then restrict.
Example Example Example (x² - 9) / (x + 3) = (x-3)(x+3) / (x+3) = x - 3, x ≠ -3
A worked simplification.

The rule:

\[\dfrac{a\cdot c}{b\cdot c}=\dfrac{a}{b}\quad(c\neq0)\]
cancel a common factor from top and bottom
Keep the excluded values from the cancelled factor.

How to simplify

  1. Factor everything.
  2. Cancel common factors.
  3. Note the excluded values.
  4. Write the reduced expression with its restrictions.
Example 1 — Difference of squares
Simplify \(\dfrac{x^2-9}{x+3}\).
Solution

Factor and cancel.

\(\dfrac{(x-3)(x+3)}{x+3}\)\(=\)\(x-3,\ x\neq-3\)
simplifies to x minus 3, with x not equal to negative 3
Example 2 — Two trinomials
Simplify \(\dfrac{x^2+5x+6}{x^2-4}\).
Solution

Factor both.

\(\dfrac{(x+2)(x+3)}{(x-2)(x+2)}\)\(=\)\(\dfrac{x+3}{x-2}\)
simplifies to x plus 3 over x minus 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.

\(\dfrac{2(x-2)(x+2)}{(x-2)^2}\)\(=\)\(\dfrac{2(x+2)}{x-2}\)
simplifies to 2 times x plus 2 over x minus 2
Example 4 — State the restriction
What restriction applies to the answer in Example 1?
Solution

The cancelled factor still restricts the domain.

\(x\)\(\neq\)\(-3\)
x cannot equal negative 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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