Lesson 3: Multiplying and Dividing Rational Expressions

Property

A rational expression is considered simplified if there are no common factors in its numerator and denominator. According to the Equivalent Fractions Property, if $a$, $b$, and $c$ are numbers where $b \neq 0$, $c \neq 0$, then $\frac{a}{b} = \frac{a \cdot c}{b \cdot c}$ and $\frac{a \cdot c}{b \cdot c} = \frac{a}{b}$. To simplify, factor the numerator and denominator completely, then divide out any common factors.

Examples

• To simplify $\frac{x^2+6x+9}{x^2-9}$, factor it as $\frac{(x+3)(x+3)}{(x-3)(x+3)}$. After canceling the common factor $(x+3)$, the simplified form is $\frac{x+3}{x-3}$.
• To simplify $\frac{5a-15}{a^2-3a}$, factor it as $\frac{5(a-3)}{a(a-3)}$. Cancel the common factor $(a-3)$ to get $\frac{5}{a}$.
• To simplify $\frac{2y^2+4y-30}{3y-9}$, factor it as $\frac{2(y+5)(y-3)}{3(y-3)}$. Cancel the common factor $(y-3)$ to get $\frac{2(y+5)}{3}$.

Explanation

Simplifying a rational expression is like reducing a fraction to its simplest form. You factor both the numerator and denominator, then cancel out any identical factors that appear on both top and bottom. Remember, only factors can be canceled, not terms!

Property

If $p$, $q$, $r$, and $s$ are polynomials where $q \neq 0$ and $s \neq 0$, then:

Examples

• To multiply $\frac{4x}{x-2} \cdot \frac{x^2-4}{8x^2}$, factor to get $\frac{4x}{x-2} \cdot \frac{(x-2)(x+2)}{8x^2}$. Cancel common factors to get $\frac{x+2}{2x}$.
• To multiply $\frac{a^2+a-6}{a^2-9} \cdot \frac{a-3}{a-2}$, factor to get $\frac{(a+3)(a-2)}{(a-3)(a+3)} \cdot \frac{a-3}{a-2}$. All factors cancel, so the result is $1$.
• To multiply $\frac{10n}{n^2+7n+10} \cdot \frac{n+2}{20n^2}$, factor to get $\frac{10n}{(n+2)(n+5)} \cdot \frac{n+2}{20n^2}$. After canceling, the result is $\frac{1}{2n(n+5)}$.

Explanation

To multiply rational expressions, factor everything you can first. Then, multiply the numerators together and the denominators together. Finally, cancel any common factors from the top and bottom. Factoring first makes finding common factors much easier.