# 7.3 – Simplifying Rational Expressions

## Review

Reducing Fractions to Simplest Form
• Step 1: To simplify a fraction, factor the its numerator and denominator.
• Step 2: Look for the greatest common factor (GCF) and cancel it.
• Ex. $\frac{6}{24}$ can be canceled by dividing the numerator and denominator by $\frac{6}{6}$
• So, $\frac{6}{24}\div \frac{6}{6}=\frac{1}{4}$

## Notes

Rational Expressions
• The first step in simplifying a rational expression is to factor its numerator and denominator.
• Then cancel those factors by dividing the numerator and denominator by the GCF.
• Example of a rational expression: $\frac{x^{2}-5x-14}{x-7}$

GCF – Greatest Common Factor
• The GCF of a rational expression is the largest factors that can be canceled from both the numerator and denominator.
• Ex. $\frac{3x-15}{x^{2}-x-20}$
• Factor the numerator and denominator: Ex. $\frac{3(x-5)}{(x-5)(x+4)}$
• Since the numerator and denominator both have the factor: (x-5), this is the GCF.

## Examples

 Ex 1. Which of the following is equal to the rational expression when x ≠ 1 or -1? $\frac{5(x-1)}{(x+1)(x-1)}$ Note: The denominator cannot equal 0, which is why x ≠ 1 and x ≠ -1 Cancel common factors from the numerator and denominator. Divide the numerator and denominator by $\frac{(x-1)}{(x-1)}$ and they will cancel. Answer: $\frac{5}{x+1}$ Ex 2. Which of the following is equal to the rational expression when x ≠ -6? $\frac{x^{2}-36}{x+6}$ Note: The denominator cannot equal 0, which is why x ≠ -6 First, factor the numerator into: $(x+6)(x-6)$ Divide the numerator and denominator by $\frac{(x+6)}{(x+6)}$ and they will cancel. Answer: $x-6$ Ex 3. Which of the following is equal to the rational expression when x ≠ 5 or x ≠ -3? $\frac{x^{2}+x-6}{x^{2}-2x-15}$ Factor the numerator and denominator and look for factors to cancel. $\frac{(x+3)(x-2)}{(x-5)(x+3)}$ Cancel the (x+3) factors. Answer: $\frac{(x-2)}{(x-5)}$ Ex 4. A rational expression has been simplified below. $\frac{(x+4)(x-2)}{9(x-2)}=\frac{x+4}{9}$ For what values of x are the two expressions equal? First, ask yourself, “What would make the denominator undefined?” Second, cancel any factors that exist in the numerator and denominator. The answer is “All real numbers except the number(s) that make the denominator undefined.” For this problem, $x-2=0$ is in the denominator, so 2 would make the epxression undefined. Answer: All real numbers except 2 Ex 5. What is the simplified form of the rational expression below? $\frac{6x^{2}-54}{5x^{2}+15x}$ Factor the numerator and denominator: $\frac{6(x+3)(x-3)}{5x(x+3)}$ The GCF is (x+3), so cancel it. Answer: $\frac{6(x-3)}{5x}$