# 7.1 Multiply and Divide Rational Expressions

Topics covered in this section are:

We previously reviewed the properties of fractions and their operations. We introduced rational numbers, which are just fractions where the numerators and denominators are integers. In this chapter, we will work with fractions whose numerators and denominators are polynomials. We call this kind of expression a rational expression.

### RATIONAL EXPRESSION

A rational expression is an expression of the form$\frac{p}{q}$, where $p$ and $q$ are polynomials and $q≠0$.

Here are some examples of rational expressions:

 $-\frac{24}{56}$ $\frac{5x}{12y}$ $\frac{4x+1}{x^{2}-9}$ $\frac{4x^{2}+3x-1}{2x-8}$

Notice that the first rational expression listed above, $-\frac{24}{56}$, is just a fraction. Since a constant is a polynomial with degree zero, the ratio of two constants is a rational expression, provided the denominator is not zero.

We will do the same operations with rational expressions that we did with fractions. We will simplify, add, subtract, multiply, divide and use them in applications.

## 7.1.1 Determine the Values for Which a Rational Expression is Undefined

If the denominator is zero, the rational expression is undefined. The numerator of a rational expression may be $0$—but not the denominator.

When we work with a numerical fraction, it is easy to avoid dividing by zero because we can see the number in the denominator. In order to avoid dividing by zero in a rational expression, we must not allow values of the variable that will make the denominator be zero.

So before we begin any operation with a rational expression, we examine it first to find the values that would make the denominator zero. That way, when we solve a rational equation for example, we will know whether the algebraic solutions we find are allowed or not.

### HOW TO: Determine the values for which a rational expression is undefined.

1. Set the denominator equal to zero.
2. Solve the equation.

#### Example 1

Determine the value for which each rational expression is undefined:

• $\frac{8a^{2}b}{3c}$
• $\frac{4b-3}{2b+5}$
• $\frac{x+4}{x^{2}+5x+6}$
Solution

Part 1

Part 2

Part 3

## 7.1.2 Simplify Rational Expressions

A fraction is considered simplified if there are no common factors, other than $1$, in its numerator and denominator. Similarly, a simplified rational expression has no common factors, other than $1$, in its numerator and denominator.

### SIMPLIFIED RATIONAL EXPRESSION

A rational expression is considered simplified if there are no common factors in its numerator and denominator.

For example,

 $\frac{x+2}{x+3}$ is simplified because there are no common factors of $x+2$ and $x+3$. $\frac{2x}{3x}$ is not simplified because $x$ is a common factor of $2x$ and $3x$.

We use the Equivalent Fractions Property to simplify numerical fractions. We restate it here as we will also use it to simplify rational expressions.

### EQUIVALENT FRACTIONS PROPERTY

If $a$, $b$, and $c$ are numbers where $b≠0$, $c≠0$, then

$\frac{a}{b} = \frac{a \cdot c}{b \cdot c}$ and $\frac{a \cdot c}{b \cdot c}=\frac{a}{b}$.

Notice that in the Equivalent Fractions Property, the values that would make the denominators zero are specifically disallowed. We see $b≠0$, $c≠0$ clearly stated.

To simplify rational expressions, we first write the numerator and denominator in factored form. Then we remove the common factors using the Equivalent Fractions Property.

Be very careful as you remove common factors. Factors are multiplied to make a product. You can remove a factor from a product. You cannot remove a term from a sum.

#### Example 2

Simplify: $\frac{x^{2}+5x+6}{x^{2}+8x+12}$.

Solution

### HOW TO: Divide rational expressions.

1. Rewrite the division as the product of the first rational expression and the reciprocal of the second.
2. Factor the numerators and denominators completely.
3. Multiply the numerators and denominators together.
4. Simplify by dividing out common factors.

Recall from Use the Language of Algebra that a complex fraction is a fraction that contains a fraction in the numerator, the denominator or both. Also, remember a fraction bar means division. A complex fraction is another way of writing division of two fractions.

#### Example 8

Divide: $\frac{ \frac{6x^{2}-7x+2}{4x-8}} { \frac{2x^{2}-7x+3}{x^{2}-5x+6}}$.

Solution

 $\frac{ \frac{6x^{2}-7x+2}{4x-8}} { \frac{2x^{2}-7x+3}{x^{2}-5x+6}}$ Rewrite with a division sign. $\frac{6x^{2}-7x+2}{4x-8} \div \frac{2x^{2}-7x+3}{x^{2}-5x+6}$ Rewrite as product of first times reciprocal of second. $\frac{6x^{2}-7x+2}{4x-8} \cdot \frac{x^{2}-5x+6}{2x^{2}-7x+3}$ Factor the numerators and the denominators, and then multiply. $\frac{(2x-1)(3x-2)(x-2)(x-3)}{4(x-2)(2x-1)(x-3)}$ Simplify by dividing out common factors. $\frac{\cancel{(2x-1)}(3x-2)\cancel{(x-2)}\cancel{(x-3)}}{4\cancel{(x-2)}\cancel{(2x-1)}\cancel{(x-3)}}$ Simplify. $\frac{3x-2}{4}$

If we have more than two rational expressions to work with, we still follow the same procedure. The first step will be to rewrite any division as multiplication by the reciprocal. Then, we factor and multiply.

#### Example 9

Perform the indicated operations: $\frac{3x-6}{4x-4} \cdot \frac{x^{2}+2x-3}{x^{2}-3x-10} \div \frac{2x+12}{8x+16}$.

Solution

 $\frac{3x-6}{4x-4} \cdot \frac{x^{2}+2x-3}{x^{2}-3x-10} \div \frac{2x+12}{8x+16}$ Rewrite the division as multiplication by the reciprocal. $\frac{3x-6}{4x-4} \cdot \frac{x^{2}+2x-3}{x^{2}-3x-10} \cdot \textcolor{red}{\frac{8x+16}{2x+12}}$ Factor the numerators and denominators. $\frac{3(x-2)}{4(x-1)} \cdot \frac{(x+3)(x-1)}{(x+2)(x-5)} \cdot \frac{8(x+2)}{2(x+6)}$ Multiply the fractions. Bringing the constants to the front will help when removing common factors. $\frac{3 \cdot 8(x-2)(x+3)(x-1)(x+2)}{4 \cdot 2(x-1)(x+2)(x-5)(x+6)}$ Simplify by dividing out common factors. $\frac{3 \cdot \cancel{8}(x-2)(x+3)\cancel{(x-1)}\cancel{(x+2)}}{\cancel{4} \cdot \cancel{2}\cancel{(x-1)}\cancel{(x+2)}(x-5)(x+6)}$ Simplify. $\frac{3(x-2)(x+3)}{(x-5)(x+6)}$

## 7.1.5 Multiply and Divide Rational Functions

We started this section stating that a rational expression is an expression of the form $pq$, $p$ and $q$ are polynomials and $q≠0$. Similarly, we define a rational function as 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.

### RATIONAL FUNCTION

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 would cause division by zero. We must eliminate any values that make $q(x)=0$.

### HOW TO: Determine the domain of a rational function.

1. Set the denominator equal to zero.
2. Solve the equation.
3. The domain is all real numbers excluding the values found in Step 2.

#### Example 10

Find the domain of $R(x) = \frac{2x^{2}-14x}{4x^{2}-16x-48}$.

Solution

The domain will be all real numbers except those values that make the denominator zero. We will set the denominator equal to zero, solve that equation, and then exclude those values from the domain.

 Set the denominator to zero. $4x^{2}-16x-48=0$ Factor, first factor out the GCF. $4(x^{2}-4x-12)=0$ $4(x-6)(x+2)=0$ Use the Zero Product Property. $4≠0 \ \ \ x-6=0 \ \ \ x+2 = 0$ Solve. $x=6 \ \ \ \ \ \ \ x=-2$ The domain of $R(x)$ is all real numbers where $x≠6$ and $x≠-2$.

To multiply rational functions, we multiply the resulting rational expressions on the right side of the equation using the same techniques we used to multiply rational expressions.

#### Example 11

Find $R(x) = f(x) \cdot g(x)$ where $f(x)=\frac{2x-6}{x^{2}-8x+15}$ and $g(x)=\frac{x^{2}-25}{2x+10}$.

Solution
 $R(x) = f(x) \cdot g(x)$ $R(x)=\frac{2x-6}{x^{2}-8x+15} \cdot \frac{x^{2}-25}{2x+10}$ Factor each numerator and denominator. $R(x)=\frac{2(x-3)}{(x-3)(x+5)} \cdot \frac{(x-5)(x+5)}{2(x+5)}$ Multiply the numerators and denominators. $R(x)=\frac{2(x-3)(x-5)(x+5)}{2(x-3)(x-5)(x+5)}$ Remove common factors. $R(x)=\frac{\cancel{2}\cancel{(x-3)}\cancel{(x-5)}\cancel{(x+5)}}{\cancel{2}\cancel{(x-3)}\cancel{(x-5)}\cancel{(x+5)}}$ Simplify. $R(x)=1$

To divide rational functions, we divide the resulting rational expressions on the right side of the equation using the same techniques we used to divide rational expressions.

#### Example 12

Find $R(x) = \frac{f(x)}{g(x)}$ where $f(x)=\frac{3x^{2}}{x^{2}-4x}$ and $g(x)=\frac{9x^{2}-45x}{x^{2}-7x+10}$.

Solution

 $R(x)=\frac{f(x)}{g(x)}$ Substitute in the functions $f(x)$, $g(x)$. $R(x)=\frac{ \frac{3x^{2}}{x^{2}-4x} }{\frac{9x^{2}-45x}{x^{2}-7x+10}}$ Rewrite the division as the product of $f(x)$ and the reciprocal of $g(x)$. $R(x)= \frac{3x^{2}}{x^{2}-4x} \cdot \frac{x^{2}-7x+10}{9x^{2}-45x}$ Factor the numerators and denominators and then multiply. $R(x)=\frac{3 \cdot x \cdot x (x-5)(x-2)} {x(x-4)\cdot 3 \cdot 3 \cdot x(x-5)}$ Simplify by dividing out common factors. $R(x)=\frac{\cancel{3} \cdot \cancel{x} \cdot \cancel{x} \cancel{(x-5)}(x-2)} {\cancel{x}(x-4)\cdot \cancel{3} \cdot 3 \cdot \cancel{x}\cancel{(x-5)}}$ $R(x)=\frac{x-2}{3(x-4)}$