Lesson 3: Multiplying and Dividing Rational Expressions

Property

A rational expression is an expression of the form $\frac{p}{q}$, where $p$ and $q$ are polynomials and $q \neq 0$. To determine the values for which a rational expression is undefined, set the denominator equal to zero and solve the equation. The expression is undefined for any value of the variable that makes the denominator zero.

Examples

• The expression $\frac{10x}{y-3}$ is undefined when $y-3=0$, so it is undefined for $y=3$.
• The expression $\frac{5a+1}{4a+2}$ is undefined when $4a+2=0$, which means $a = -\frac{1}{2}$.
• The expression $\frac{m-5}{m^2-m-12}$ is undefined when $m^2-m-12=0$. Factoring gives $(m-4)(m+3)=0$, so it is undefined for $m=4$ or $m=-3$.

Explanation

Think of a rational expression as a fraction with polynomials. Just like you can't divide by zero in arithmetic, you can't have a zero in the denominator of a rational expression. Finding these 'forbidden' values is a crucial first step.

Property

1. Factor numerator and denominator completely.
2. Divide numerator and denominator by any common factors. We can cancel common factors (expressions that are multiplied together), but not common terms (expressions that are added or subtracted).

Examples

Problem: Reduce the fraction $\frac{x^2 + 3x + 2}{x^2 - 4}$.

Step 1: Factor everything.
The numerator factors into $(x+1)(x+2)$.
The denominator is a difference of squares, factoring into $(x-2)(x+2)$.

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!