Lesson 7.1: Multiply and Divide Rational Expressions

New Concept

This lesson introduces rational expressions—fractions with polynomials. You'll learn to simplify, multiply, and divide them by factoring, similar to regular fractions. A key first step is always finding values that make the denominator zero, which makes the expression undefined.

What’s next

You'll start by determining when expressions are undefined. Then, you'll master simplifying, multiplying, and dividing them through interactive examples and a series of practice cards.

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

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

The opposite of $a-b$ is $b-a$. An expression and its opposite divide to $-1$. Therefore, for any expression where $a \neq b$:

Examples

• To simplify $\frac{z^2-25}{5-z}$, factor the numerator to get $\frac{(z-5)(z+5)}{5-z}$. Since $\frac{z-5}{5-z} = -1$, the expression simplifies to $-(z+5)$.
• To simplify $\frac{9-x^2}{x^2-x-6}$, factor both parts to get $\frac{(3-x)(3+x)}{(x-3)(x+2)}$. The factors $(3-x)$ and $(x-3)$ are opposites, so their ratio is $-1$. The result is $-\frac{x+3}{x+2}$.
• The expression $\frac{y-2}{2-y}$ simplifies directly to $-1$, as the numerator and denominator are opposites.

Explanation

When you spot factors that are subtracted in reverse order, like $(x-4)$ and $(4-x)$, they are opposites. These pairs don't cancel to 1; they cancel to $-1$. This is a handy trick for simplifying certain expressions.

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.

Property

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

Examples

• To divide $\frac{x^2-y^2}{7x} \div \frac{x+y}{21y}$, rewrite it as $\frac{(x-y)(x+y)}{7x} \cdot \frac{21y}{x+y}$. After canceling, the result is $\frac{3y(x-y)}{x}$.
• To divide $\frac{a^2-4a+3}{a^2-4} \div \frac{a-1}{a+2}$, rewrite it as $\frac{(a-3)(a-1)}{(a-2)(a+2)} \cdot \frac{a+2}{a-1}$. The simplified result is $\frac{a-3}{a-2}$.
• To divide $\frac{x^3-27}{x^2-1} \div \frac{x^2+3x+9}{x-1}$, rewrite as $\frac{(x-3)(x^2+3x+9)}{(x-1)(x+1)} \cdot \frac{x-1}{x^2+3x+9}$. The simplified result is $\frac{x-3}{x+1}$.

Explanation

Dividing rational expressions is just like dividing fractions: 'Keep, Change, Flip.' Keep the first expression the same, change the division sign to multiplication, and flip the second expression (use its reciprocal). Then, simply multiply and simplify.

Property

A rational function is a function of the form $R(x) = \frac{p(x)}{q(x)}$ where $p(x)$ and $q(x)$ are polynomial functions and $q(x)$ is not zero. The domain of a rational function is all real numbers except for those values that make the denominator $q(x) = 0$. To multiply or divide rational functions, you perform the operation on their corresponding rational expressions.

Examples

• Find the domain of $R(x) = \frac{3x-5}{x^2-16}$. Set $x^2-16=0$, which gives $x=4$ and $x=-4$. The domain is all real numbers except $4$ and $-4$.
• If $f(x) = \frac{x-5}{2x}$ and $g(x) = \frac{x^2-25}{6x}$, find $R(x) = f(x) \cdot g(x)$. This is $\frac{x-5}{2x} \cdot \frac{(x-5)(x+5)}{6x} = \frac{(x-5)^2(x+5)}{12x^2}$.
• If $f(x) = \frac{x+2}{x^2-9}$ and $g(x) = \frac{x+2}{x-3}$, find $R(x) = \frac{f(x)}{g(x)}$. This is $\frac{x+2}{(x-3)(x+3)} \cdot \frac{x-3}{x+2}$. After canceling, $R(x) = \frac{1}{x+3}$.

Explanation

A rational function is just a function defined by a fraction of polynomials. To find its domain, find all x-values that make the denominator zero and exclude them. To multiply or divide the functions, just multiply or divide their expressions.