### Learning Objectives

In this section students will:

A pastry shop has fixed costs of $\,\text{\}280\,$ per week and variable costs of $\,\text{\}9\,$ per box of pastries. The shop’s costs per week in terms of $\,x,$ the number of boxes made, is $\,280+9x.\,$ We can divide the costs per week by the number of boxes made to determine the cost per box of pastries.

$\frac{280+9x}{x}$

Notice that the result is a polynomial expression divided by a second polynomial expression. In this section, we will explore quotients of polynomial expressions.

### 6.1.1 – Simplifying Rational Expressions

The quotient of two polynomial expressions is called a rational expression. We can apply the properties of fractions to rational expressions, such as simplifying the expressions by canceling common factors from the numerator and the denominator. To do this, we first need to factor both the numerator and denominator. Let’s start with the rational expression shown.

$\frac{{x}^{2}+8x+16}{{x}^{2}+11x+28}$

We can factor the numerator and denominator to rewrite the expression.

$\frac{{\left(x+4\right)}^{2}}{\left(x+4\right)\left(x+7\right)}$

Then we can simplify that expression by canceling the common factor $\,\left(x+4\right).$

$\frac{x+4}{x+7}$

### How To

Given a rational expression, simplify it.

1. Factor the numerator and denominator.
2. Cancel any common factors.

### Example 1 – Simplifying Rational Expressions

Simplify $\,\frac{{x}^{2}-9}{{x}^{2}+4x+3}.$

$\begin{array}{lllll}\frac{\left(x+3\right)\left(x-3\right)}{\left(x+3\right)\left(x+1\right)}\hfill & \hfill & \hfill & \hfill & \text{Factor the numerator and the denominator}.\hfill \\ \frac{x-3}{x+1}\hfill & \hfill & \hfill & \hfill & \text{Cancel common factor }\left(x+3\right).\hfill \end{array}$

#### Analysis

We can cancel the common factor because any expression divided by itself is equal to 1.

Can the $\,{x}^{2}\,$ term be cancelled in Example 1?

No. A factor is an expression that is multiplied by another expression. The $\,{x}^{2}\,$ term is not a factor of the numerator or the denominator.

### Try It

Simplify $\,\frac{x-6}{{x}^{2}-36}.$

$\frac{1}{x+6}$

### 6.1.2 – Multiplying Rational Expressions

Multiplication of rational expressions works the same way as multiplication of any other fractions. We multiply the numerators to find the numerator of the product, and then multiply the denominators to find the denominator of the product. Before multiplying, it is helpful to factor the numerators and denominators just as we did when simplifying rational expressions. We are often able to simplify the product of rational expressions.

### How To

Given two rational expressions, multiply them.

1. Factor the numerator and denominator.
2. Multiply the numerators.
3. Multiply the denominators.
4. Simplify.

### Example 2 – Multiplying Rational Expressions

Multiply the rational expressions and show the product in simplest form:

$\frac{\left(x+5\right)\left(x-1\right)}{3\left(x+6\right)}\cdot \frac{\left(2x-1\right)}{\left(x+5\right)}$
$\begin{array}{lllll}\frac{\left(x+5\right)\left(x-1\right)}{3\left(x+6\right)}\cdot \frac{\left(2x-1\right)}{\left(x+5\right)}\hfill & \hfill & \hfill & \hfill & \text{Factor the numerator and denominator}.\hfill \\ \frac{\left(x+5\right)\left(x-1\right)\left(2x-1\right)}{3\left(x+6\right)\left(x+5\right)}\hfill & \hfill & \hfill & \hfill & \text{Multiply numerators and denominators}.\hfill \\ \frac{\cancel{\left(x+5\right)}\left(x-1\right)\left(2x-1\right)}{3\left(x+6\right)\cancel{\left(x+5\right)}}\hfill & \hfill & \hfill & \hfill & \text{Cancel common factors to simplify}.\hfill \\ \frac{\left(x-1\right)\left(2x-1\right)}{3\left(x+6\right)} \hfill & \hfill & \hfill & \hfill & \hfill \end{array}$

### Try It

Multiply the rational expressions and show the product in simplest form:

$\frac{{x}^{2}+11x+30}{{x}^{2}+5x+6}\cdot \frac{{x}^{2}+7x+12}{{x}^{2}+8x+16}$

$\frac{\left(x+5\right)\left(x+6\right)}{\left(x+2\right)\left(x+4\right)}$

### 6.1.3 – Dividing Rational Expressions

Division of rational expressions works the same way as division of other fractions. To divide a rational expression by another rational expression, multiply the first expression by the reciprocal of the second. Using this approach, we would rewrite $\,\frac{1}{x}÷\frac{{x}^{2}}{3}\,$ as the product $\,\frac{1}{x}\cdot \frac{3}{{x}^{2}}.\,$ Once the division expression has been rewritten as a multiplication expression, we can multiply as we did before.

$\frac{1}{x}\cdot \frac{3}{{x}^{2}}=\frac{3}{{x}^{3}}$

### How To

Given two rational expressions, divide them.

1. Rewrite as the first rational expression multiplied by the reciprocal of the second.
2. Factor the numerators and denominators.
3. Multiply the numerators.
4. Multiply the denominators.
5. Simplify.

### Example 3 – Dividing Rational Expressions

Divide the rational expressions and express the quotient in simplest form:

$\frac{2{x}^{2}+x-6}{{x}^{2}-1}\div\frac{{x}^{2}-4}{{x}^{2}+2x+1}$
$\begin{array}{lllll}\frac{2{x}^{2}+x-6}{{x}^{2}-1}\cdot\frac{{x}^{2}+2x+1}{{x}^{2}-4}\hfill & \hfill & \hfill & \hfill & \text{Rewrite using multiplication}.\hfill \\ \frac{(2x-3)(x+2)}{(x+1)(x-1)}\cdot \frac{(x+1)(x+1)}{(x+2)(x-2)}\hfill & \hfill & \hfill & \hfill & \text{Factor the numerator and denominator}.\hfill \\ \frac{(2x-3)(x+2)(x+1)(x+1)}{(x+1)(x-1)(x+2)(x-2)}\hfill & \hfill & \hfill & \hfill & \text{Multiply numerators and denominators}.\hfill \\ \frac{(2x-3)\cancel{(x+2)}\cancel{(x+1)}(x+1)}{\cancel{(x+1)}(x-1)\cancel{(x+2)}(x-2)}\hfill & \hfill & \hfill & \hfill & \text{Cancel common factors to simplify}.\hfill \\ \frac{(2x-3)(x+1)}{(x-1)(x-2)} \hfill & \hfill & \hfill & \hfill & \hfill \end{array}$

### Try It

Divide the rational expressions and express the quotient in simplest form:

$\frac{9{x}^{2}-16}{3{x}^{2}+17x-28}\div\frac{3{x}^{2}-2x-8}{{x}^{2}+5x-14}$

$1$

Access these online resources for additional instruction and practice with rational expressions.

### Key Concepts

• Rational expressions can be simplified by cancelling common factors in the numerator and denominator. See Example 1.
• We can multiply rational expressions by multiplying the numerators and multiplying the denominators. See Example 2.
• To divide rational expressions, multiply by the reciprocal of the second expression. See Example 3.

### Glossary

rational expression
the quotient of two polynomial expressions