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

[latex]\frac{280+9x}{x}[/latex]

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

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.

[latex]\frac{{x}^{2}+8x+16}{{x}^{2}+11x+28}[/latex]

[latex]\\[/latex]We can factor the numerator and denominator to rewrite the expression as [latex]\frac{{\left(x+4\right)}^{2}}{\left(x+4\right)\left(x+7\right)}[/latex]

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.

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 [latex]\frac{1}{x}\div \frac{{x}^{2}}{3}[/latex] as the product [latex]\frac{1}{x}\cdot \frac{3}{{x}^{2}}[/latex]. Once the division expression has been rewritten as a multiplication expression, we can multiply as we did before.

Adding and Subtracting Rational Expressions

Adding and subtracting rational expressions works just like adding and subtracting numerical fractions. To add fractions, we need to find a common denominator. Let’s look at an example of fraction addition.

We have to rewrite the fractions so they share a common denominator before we are able to add. We must do the same thing when adding or subtracting rational expressions.

The easiest common denominator to use will be the least common denominator or LCD. The LCD is the smallest multiple that the denominators have in common. To find the LCD of two rational expressions, we factor the expressions and multiply all of the distinct factors. For instance, if the factored denominators were [latex]\left(x+3\right)\left(x+4\right)[/latex] and [latex]\left(x+4\right)\left(x+5\right)[/latex], then the LCD would be [latex]\left(x+3\right)\left(x+4\right)\left(x+5\right)[/latex].

Once we find the LCD, we need to multiply each expression by the form of 1 that will change the denominator to the LCD. We would need to multiply the expression with a denominator of [latex]\left(x+3\right)\left(x+4\right)[/latex] by [latex]\frac{x+5}{x+5}[/latex] and the expression with a denominator of [latex]\left(x+4\right)\left(x+5\right)[/latex] by [latex]\frac{x+3}{x+3}[/latex].

Simplifying Complex Rational Expressions

A complex rational expression is a rational expression that contains additional rational expressions in the numerator, the denominator, or both. We can simplify complex rational expressions by rewriting the numerator and denominator as single rational expressions and dividing. The complex rational expression [latex]\frac{a}{\frac{1}{b}+c}[/latex] can be simplified by rewriting the numerator as the fraction [latex]\frac{a}{1}[/latex] and combining the expressions in the denominator as [latex]\frac{1+bc}{b}[/latex]. We can then rewrite the expression as a multiplication problem using the reciprocal of the denominator. We get [latex]\frac{a}{1}\cdot \frac{b}{1+bc}[/latex] which is equal to [latex]\frac{ab}{1+bc}[/latex].

Key Concepts

• Rational expressions can be simplified by canceling common factors in the numerator and denominator.
• We can multiply rational expressions by multiplying the numerators and multiplying the denominators.
• To divide rational expressions, multiply by the reciprocal of the second expression.
• Adding or subtracting rational expressions requires finding a common denominator.
• Complex rational expressions have fractions in the numerator or the denominator. These expressions can be simplified.

Glossary

least common denominator

the smallest multiple that two denominators have in common

rational expression

the quotient of two polynomial expressions