Simplifying Rational Expressions Math Example 1

Follow the full solution, then compare it with the other examples linked below.

Example 1

medium
Simplify x2โˆ’9x2+5x+6\frac{x^2 - 9}{x^2 + 5x + 6}.

Solution

  1. 1
    Step 1: Factor numerator: x2โˆ’9=(x+3)(xโˆ’3)x^2 - 9 = (x+3)(x-3).
  2. 2
    Step 2: Factor denominator: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x+2)(x+3).
  3. 3
    Step 3: Cancel common factor (x+3)(x+3): (x+3)(xโˆ’3)(x+2)(x+3)=xโˆ’3x+2\frac{(x+3)(x-3)}{(x+2)(x+3)} = \frac{x-3}{x+2}, xโ‰ โˆ’3x \neq -3.
  4. 4
    Check: At x=1x = 1: 1โˆ’91+5+6=โˆ’812=โˆ’23\frac{1-9}{1+5+6} = \frac{-8}{12} = -\frac{2}{3} and 1โˆ’31+2=โˆ’23\frac{1-3}{1+2} = -\frac{2}{3} โœ“

Answer

xโˆ’3x+2\frac{x - 3}{x + 2}, xโ‰ โˆ’3x \neq -3
To simplify a rational expression, factor both numerator and denominator completely, then cancel common factors. Always note the excluded values where the original denominator was zero.

About Simplifying Rational Expressions

Simplifying a rational expression p(x)q(x)\frac{p(x)}{q(x)} by factoring both the numerator and denominator, then canceling common factors. The domain must exclude values that make any original denominator zero.

Learn more about Simplifying Rational Expressions โ†’

More Simplifying Rational Expressions Examples

Example 2 easy

Simplify [formula].

Example 3 easy

Simplify [formula].

Example 4 hard

Simplify [formula].

โ† All Simplifying Rational Expressions Examples Simplifying Rational Expressions Concept