EDGE Math 8

Simplest Form of Rational Expressions

Example 2

Simplify each rational expression.

\frac{3 b^{2} - 3 b}{3 b^{2} + 6 b}

\frac{x^{2} - 2 x - 8}{x^{2} - 3 x - 10}

Solution

a. Factor the numerator and the denominator. Then write $\frac{3 b^{2} - 3 b}{3 b^{2} + 6 b}$ as a product of two rational expressions and cancel the factor that is equal to 1.

\frac{3 b^{2} - 3 b}{3 b^{2} + 6 b} = \frac{3 b (b - 1)}{3 b (b + 2)}

= \frac{3 b}{3 b} \cdot \frac{b - 1}{b + 2}

= 1 \cdot \frac{b - 1}{b + 2}

= \frac{b - 1}{b + 2}

b. Factor the numerator and the denominator. Then write $\frac{x^{2} - 2 x - 8}{x^{2} - 3 x - 10}$ as a product of two rational expressions and cancel the factor that is equal to 1.

\frac{x^{2} - 2 x - 8}{x^{2} - 3 x - 10} = \frac{(x + 2) (x - 4)}{(x + 2) (x - 5)}

= \frac{x + 2}{x + 2} \cdot \frac{x - 4}{x - 5}

= 1 \cdot \frac{x - 4}{x - 5}

= \frac{x - 4}{x - 5}

It is important to note that when canceling factors that are equal to 1, like $\frac{x + 2}{x + 2}$ in the example above, x cannot have a value of –2 because if $x = - 2,$ the denominator becomes 0.