EDGE Math 8

Simplest Form of Rational Expressions

Note that canceling the common factors of the numerator and the denominator of a rational expression is a shortcut for simplifying a factor that is equal to 1. For instance, in the previous slide, the expression in the first example can be simplified using cancelation as follows:

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

The expression $3 b$ in both the numerator and the denominator can be canceled out because $\frac{3 b}{3 b} = 1$ when $b \neq 0.$

As a general rule, if both b and c are not equal to 0, then

\frac{a c}{b c} = \frac{a}{b} .

This rule is called the cancelation property.

When applying the cancelation property, you just need to be careful in identifying expressions that you can cancel. For example, in the expression $\frac{2 x + 3}{2 x - 5},$ you cannot cancel $2 x$ in the numerator and the denominator even if $\frac{2 x}{2 x}$ is equal to 1 because $2 x$ is not a factor of $2 x + 3$ and $2 x - 5$ In fact, $\frac{2 x + 3}{2 x - 5},$ is already in simplest form.