1.6 Rational expressions

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In this section students will:
• Simplify rational expressions.
• Multiply rational expressions.
• Divide rational expressions.
• Add and subtract rational expressions.
• Simplify complex rational expressions.

A pastry shop has fixed costs of $\text{\hspace{0.17em}}\text{}280\text{\hspace{0.17em}}$ per week and variable costs of $\text{\hspace{0.17em}}\text{}9\text{\hspace{0.17em}}$ per box of pastries. The shop’s costs per week in terms of $\text{\hspace{0.17em}}x,$ the number of boxes made, is $\text{\hspace{0.17em}}280+9x.\text{\hspace{0.17em}}$ 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.

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 $\text{\hspace{0.17em}}\left(x+4\right).$

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

Given a rational expression, simplify it.

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

Simplifying rational expressions

Simplify $\text{\hspace{0.17em}}\frac{{x}^{2}-9}{{x}^{2}+4x+3}.$

Can the $\text{\hspace{0.17em}}{x}^{2}\text{\hspace{0.17em}}$ term be cancelled in [link] ?

No. A factor is an expression that is multiplied by another expression. The $\text{\hspace{0.17em}}{x}^{2}\text{\hspace{0.17em}}$ term is not a factor of the numerator or the denominator.

Simplify $\text{\hspace{0.17em}}\frac{x-6}{{x}^{2}-36}.$

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

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.

Given two rational expressions, multiply them.

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

Multiplying rational expressions

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

$\frac{\left(x+5\right)\left(x-1\right)}{3\left(x+6\right)}\cdot \frac{\left(2x-1\right)}{\left(x+5\right)}$

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

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

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

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 $\text{\hspace{0.17em}}\frac{1}{x}÷\frac{{x}^{2}}{3}\text{\hspace{0.17em}}$ as the product $\text{\hspace{0.17em}}\frac{1}{x}\cdot \frac{3}{{x}^{2}}.\text{\hspace{0.17em}}$ Once the division expression has been rewritten as a multiplication expression, we can multiply as we did before.

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