Multiply and Divide Rational Expressions

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.

, 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.

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.

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.

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

Notice that in the Equivalent Fractions Property, the values that would make the denominators zero are specifically disallowed. We see

b \neq 0, c \neq 0

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.

Usually, we leave the simplified rational expression in factored form. This way, it is easy to check that we have removed all the common factors.

We’ll use the methods we have learned to factor the polynomials in the numerators and denominators in the following examples.

Every time we write a rational expression, we should make a statement disallowing values that would make a denominator zero. However, to let us focus on the work at hand, we will omit writing it in the examples.

Now we will see how to simplify a rational expression whose numerator and denominator have opposite factors. We previously introduced opposite notation: the opposite of a is

− a

We will use this property to simplify rational expressions that contain opposites in their numerators and denominators. Be careful not to treat

a + b

as opposites. Recall that in addition, order doesn’t matter so

a + b = b + a

Multiply Rational Expressions

To multiply rational expressions, we do just what we did with numerical fractions. We multiply the numerators and multiply the denominators. Then, if there are any common factors, we remove them to simplify the result.

Remember, throughout this chapter, we will assume that all numerical values that would make the denominator be zero are excluded. We will not write the restrictions for each rational expression, but keep in mind that the denominator can never be zero. So in this next example,

x \neq 0,

Divide Rational Expressions

Just like we did for numerical fractions, to divide rational expressions, we multiply the first fraction by the reciprocal of the second.

Once we rewrite the division as multiplication of the first expression by the reciprocal of the second, we then factor everything and look for 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.

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.

Multiply and Divide Rational Functions

We started this section stating that a rational expression is an expression of the form

\frac{p}{q},

Similarly, we define a rational function as a function of the form

R (x) = \frac{p (x)}{q (x)}

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 .

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.

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.

Key Concepts

Practice Makes Perfect

Determine the Values for Which a Rational Expression is Undefined

In the following exercises, determine the values for which the rational expression is undefined.

ⓐ $\frac{2 x^{2}}{z}$

ⓑ $\frac{4 p - 1}{6 p - 5}$

ⓐ $z = 0$

ⓑ $p = \frac{5}{6}$

ⓐ $\frac{10 m}{11 n}$

ⓑ $\frac{6 y + 13}{4 y - 9}$

ⓐ $\frac{4 x^{2} y}{3 y}$

ⓑ $\frac{3 x - 2}{2 x + 1}$

ⓐ $y = 0$

ⓑ $x = - \frac{1}{2}$

ⓐ $\frac{5 p q^{2}}{9 q}$

ⓑ $\frac{7 a - 4}{3 a + 5}$

Simplify Rational Expressions

In the following exercises, simplify each rational expression.

- \frac{44}{55}

- \frac{4}{5}

\frac{56}{63}

\frac{8 m^{3} n}{12 m n^{2}}

\frac{2 m^{2}}{3 n}

\frac{36 v^{3} w^{2}}{27 v w^{3}}

\frac{8 n - 96}{3 n - 36}

\frac{8}{3}

\frac{12 p - 240}{5 p - 100}

\frac{x^{2} + 4 x - 5}{x^{2} - 2 x + 1}

\frac{x + 5}{x - 1}

\frac{y^{2} + 3 y - 4}{y^{2} - 6 y + 5}

\frac{a^{2} - 4}{a^{2} + 6 a - 16}

\frac{a + 2}{a + 8}

\frac{y^{2} - 2 y - 3}{y^{2} - 9}

\frac{p^{3} + 3 p^{2} + 4 p + 12}{p^{2} + p - 6}

\frac{p^{2} + 4}{p - 2}

\frac{x^{3} - 2 x^{2} - 25 x + 50}{x^{2} - 25}

\frac{8 b^{2} - 32 b}{2 b^{2} - 6 b - 80}

\frac{4 b (b - 4)}{(b + 5) (b - 8)}

\frac{−5 c^{2} - 10 c}{−10 c^{2} + 30 c + 100}

\frac{3 m^{2} + 30 m n + 75 n^{2}}{4 m^{2} - 100 n^{2}}

\frac{3 (m + 5 n)}{4 (m - 5 n)}

\frac{5 r^{2} + 30 r s - 35 s^{2}}{r^{2} - 49 s^{2}}

\frac{a - 5}{5 - a}

−1

\frac{5 - d}{d - 5}

\frac{20 - 5 y}{y^{2} - 16}

- \frac{5}{y + 4}

\frac{4 v - 32}{64 - v^{2}}

\frac{w^{3} + 216}{w^{2} - 36}

\frac{w^{2} - 6 w + 36}{w - 6}

\frac{v^{3} + 125}{v^{2} - 25}

\frac{z^{2} - 9 z + 20}{16 - z^{2}}

- \frac{z - 5}{4 + z}

\frac{a^{2} - 5 z - 36}{81 - a^{2}}

Multiply Rational Expressions

In the following exercises, multiply the rational expressions.

\frac{12}{16} \cdot \frac{4}{10}

\frac{3}{10}

\frac{32}{5} \cdot \frac{16}{24}

\frac{5 x^{2} y^{4}}{12 x y^{3}} \cdot \frac{6 x^{2}}{20 y^{2}}

\frac{x^{3}}{8 y}

\frac{12 a^{3} b}{b^{2}} \cdot \frac{2 a b^{2}}{9 b^{3}}

\frac{5 p^{2}}{p^{2} - 5 p - 36} \cdot \frac{p^{2} - 16}{10 p}

\frac{p (p - 4)}{2 (p - 9)}

\frac{3 q^{2}}{q^{2} + q - 6} \cdot \frac{q^{2} - 9}{9 q}

\frac{2 y^{2} - 10 y}{y^{2} + 10 y + 25} \cdot \frac{y + 5}{6 y}

\frac{y - 5}{3 (y + 5)}

\frac{z^{2} + 3 z}{z^{2} - 3 z - 4} \cdot \frac{z - 4}{z^{2}}

\frac{28 - 4 b}{3 b - 3} \cdot \frac{b^{2} + 8 b - 9}{b^{2} - 49}

- \frac{4 (b + 9)}{3 (b + 7)}

\frac{72 m - 12 m^{2}}{8 m + 32} \cdot \frac{m^{2} + 10 m + 24}{m^{2} - 36}

\frac{5 c^{2} + 9 c + 2}{c^{2} - 25} \cdot \frac{c^{2} + 10 c + 25}{3 c^{2} - 14 c - 5}

\frac{(c + 2) (c + 2)}{(c - 2) (c - 3)}

\frac{2 d^{2} + d - 3}{d^{2} - 16} \cdot \frac{d^{2} - 8 d + 16}{2 d^{2} - 9 d - 18}

\frac{6 m^{2} - 2 m - 10}{9 - m^{2}} \cdot \frac{m^{2} - 6 m + 9}{6 m^{2} + 29 m - 20}

- \frac{(m - 2) (m - 3)}{(3 + m) (m + 4)}

\frac{2 n^{2} - 3 n - 14}{25 - n^{2}} \cdot \frac{n^{2} - 10 n + 25}{2 n^{2} - 13 n + 21}

Divide Rational Expressions

In the following exercises, divide the rational expressions.

\frac{v - 5}{11 - v} \div \frac{v^{2} - 25}{v - 11}

- \frac{1}{v + 5}

\frac{10 + w}{w - 8} \div \frac{100 - w^{2}}{8 - w}

\frac{3 s^{2}}{s^{2} - 16} \div \frac{s^{3} - 4 s^{2} + 16 s}{s^{3} - 64}

\frac{3 s}{s + 4}

\frac{r^{2} - 9}{15} \div \frac{r^{3} - 27}{5 r^{2} + 15 r + 45}

\frac{p^{3} + q^{3}}{3 p^{2} + 3 p q + 3 q^{2}} \div \frac{p^{2} - q^{2}}{12}

\frac{4 (p^{2} - p q + q^{2})}{(p - q) (p^{2} + p q + q^{2})}

\frac{v^{3} - 8 w^{3}}{2 v^{2} + 4 v w + 8 w^{2}} \div \frac{v^{2} - 4 w^{2}}{4}

\frac{x^{2} + 3 x - 10}{4 x} \div (2 x^{2} + 20 x + 50)

\frac{x - 2}{8 x}

\frac{2 y^{2} - 10 y z - 48 z^{2}}{2 y - 1} \div (4 y^{2} - 32 y z)

\frac{\frac{2 a^{2} - a - 21}{5 a + 20}}{\frac{a^{2} + 7 a + 12}{a^{2} + 8 a + 16}}

\frac{2 a - 7}{5}

\frac{\frac{3 b^{2} + 2 b - 8}{12 b + 18}}{\frac{3 b^{2} + 2 b - 8}{2 b^{2} - 7 b - 15}}

\frac{\frac{12 c^{2} - 12}{2 c^{2} - 3 c + 1}}{\frac{4 c + 4}{6 c^{2} - 13 c + 5}}

3 (3 c - 5)

\frac{\frac{4 d^{2} + 7 d - 2}{35 d + 10}}{\frac{d^{2} - 4}{7 d^{2} - 12 d - 4}}

For the following exercises, perform the indicated operations.

\frac{10 m^{2} + 80 m}{3 m - 9} \cdot \frac{m^{2} + 4 m - 21}{m^{2} - 9 m + 20} \div \frac{5 m^{2} + 10 m}{2 m - 10}

\frac{4 (m + 8) (m + 7)}{3 (m - 4) (m + 2)}

\frac{4 n^{2} + 32 n}{3 n + 2} \cdot \frac{3 n^{2} - n - 2}{n^{2} + n - 30} \div \frac{108 n^{2} - 24 n}{n + 6}

\frac{12 p^{2} + 3 p}{p + 3} \div \frac{p^{2} + 2 p - 63}{p^{2} - p - 12} \cdot \frac{p - 7}{9 p^{3} - 9 p^{2}}

\frac{(4 p + 1) (p - 7)}{3 p (p + 9) (p - 1)}

\frac{6 q + 3}{9 q^{2} - 9 q} \div \frac{q^{2} + 14 q + 33}{q^{2} + 4 q - 5} \cdot \frac{4 q^{2} + 12 q}{12 q + 6}

Multiply and Divide Rational Functions

In the following exercises, find the domain of each function.

R (x) = \frac{x^{3} - 2 x^{2} - 25 x + 50}{x^{2} - 25}

x \neq 5

and $x \neq − 5$

R (x) = \frac{x^{3} + 3 x^{2} - 4 x - 12}{x^{2} - 4}

R (x) = \frac{3 x^{2} + 15 x}{6 x^{2} + 6 x - 36}

x \neq 2

and $x \neq − 3$

R (x) = \frac{8 x^{2} - 32 x}{2 x^{2} - 6 x - 80}

For the following exercises, find $R (x) = f (x) \cdot g (x)$

where $f (x)$

and $g (x)$

are given.

f (x) = \frac{6 x^{2} - 12 x}{x^{2} + 7 x - 18}

g (x) = \frac{x^{2} - 81}{3 x^{2} - 27 x}

R (x) = 2

f (x) = \frac{x^{2} - 2 x}{x^{2} + 6 x - 16}

g (x) = \frac{x^{2} - 64}{x^{2} - 8 x}

f (x) = \frac{4 x}{x^{2} - 3 x - 10}

g (x) = \frac{x^{2} - 25}{8 x^{2}}

R (x) = \frac{x + 5}{2 x (x + 2)}

f (x) = \frac{2 x^{2} + 8 x}{x^{2} - 9 x + 20}

g (x) = \frac{x - 5}{x^{2}}

For the following exercises, find $R (x) = \frac{f (x)}{g (x)}$

where $f (x)$

and $g (x)$

are given.

f (x) = \frac{27 x^{2}}{3 x - 21}

g (x) = \frac{3 x^{2} + 18 x}{x^{2} + 13 x + 42}

R (x) = \frac{3 x (x + 7)}{x - 7}

f (x) = \frac{24 x^{2}}{2 x - 8}

g (x) = \frac{4 x^{3} + 28 x^{2}}{x^{2} + 11 x + 28}

f (x) = \frac{16 x^{2}}{4 x + 36}

g (x) = \frac{4 x^{2} - 24 x}{x^{2} + 4 x - 45}

R (x) = \frac{x (x - 5)}{x - 6}

f (x) = \frac{24 x^{2}}{2 x - 4}

g (x) = \frac{12 x^{2} + 36 x}{x^{2} - 11 x + 18}

Writing Exercises

Explain how you find the values of x for which the rational expression $\frac{x^{2} - x - 20}{x^{2} - 4}$

is undefined.

Answers will vary.

Explain all the steps you take to simplify the rational expression $\frac{p^{2} + 4 p - 21}{9 - p^{2}} .$

ⓐ Multiply $\frac{7}{4} \cdot \frac{9}{10}$

and explain all your steps.* * *

ⓑ Multiply $\frac{n}{n - 3} \cdot \frac{9}{n + 3}$

and explain all your steps.* * *

. Did you get the same answer you got in part ⓐ? Why or why not?

Answers will vary.

ⓐ Divide $\frac{24}{5} \div 6$

and explain all your steps.* * *

ⓑ Divide $\frac{x^{2} - 1}{x} \div (x + 1)$

and explain all your steps.* * *

Did you get the same answer you got in part ⓐ? Why or why not?

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ If most of your checks were:

…confidently. Congratulations! You have achieved your goals in this section! Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific!

…with some help. This must be addressed quickly as topics you do not master become potholes in your road to success. Math is sequential - every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no - I don’t get it! This is critical and you must not ignore it. You need to get help immediately or you will quickly be overwhelmed. See your instructor as soon as possible to discuss your situation. Together you can come up with a plan to get you the help you need.


Rewrite the division as multiplication by the reciprocal.
Factor the numerators and the denominators.
Multiply the fractions. Bringing the constants to the front will help when removing common factors.
Simplify by dividing out common factors.
Simplify.

Multiply and Divide Rational Expressions

Determine the Values for Which a Rational Expression is Undefined

Simplify Rational Expressions

Multiply Rational Expressions

Divide Rational Expressions

Multiply and Divide Rational Functions

Key Concepts

Practice Makes Perfect

Writing Exercises

Self Check

Glossary