4A.6: Multiply and Divide Rational Expressions

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Learning Objectives

By the end of this section, you will be able to:

Determine the values for which a rational expression is undefined
Simplify rational expressions
Multiply rational expressions
Divide rational expressions
Perform multiple operations on rational expressions

Section 10.B reviews the properties of fractions and their operations. Section 0.2 introduces rational numbers, which are just fractions where the numerators and denominators are integers. In this section, we will work with fractions whose numerators and denominators are polynomials. We call this kind of expression a rational expression.

definition: RATIONAL EXPRESSION

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

Here are some examples of rational expressions:

\[−\dfrac{24}{56} \hskip{0.6in} \dfrac{5x}{12y} \hskip{0.6in} \dfrac{4x+1}{x^2−9} \hskip{0.6in} \dfrac{4x^2+3x−1}{2x−8}\nonumber\]

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, and divide. We will also 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 must examine it 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 valid or not.

$how-to.png$ DETERMINE THE variable VALUES FOR WHICH A RATIONAL EXPRESSION IS UNDEFINED

Set the denominator equal to zero.
Solve the equation. The expression is undefined for all solutions to the equation.

Example $\PageIndex{1}$

Determine the value for which each rational expression is undefined:

a. $\dfrac{8a^2b}{3c}$ b. $\dfrac{4b−3}{2b+5}$ c. $\dfrac{x+4}{x^2+5x+6}$.

Solution

The expression will be undefined when the denominator is zero.

$\begin{array} {ll} &\dfrac{8a^2b}{3c} \\ \begin{array} {l} \text{Set the denominator equal to zero and solve} \\ \text{for the variable.} \end{array} &3c=0 \\ &c=0 \\ &\dfrac{8a^2b}{3c}\text{ is undefined for }c=0. \end{array} $
b.

$\begin{array} {ll} &\dfrac{4b-3}{2b+5} \\ \begin{array} {l} \text{Set the denominator equal to zero and solve} \\ \text{for the variable.} \end{array} &2b+5=0 \\ &2b=-5 \\ &b=-\dfrac{5}{2} \\ & \\ &\dfrac{4b-3}{2b+5} \text{ is undefined for }b=-\dfrac{5}{2}. \end{array} $

$\begin{array} {ll} &\dfrac{x+4}{x^2 + 5x + 6} \\ \begin{array} {l} \text{Set the denominator equal to zero and solve } \\ \text{for the variable.} \end{array} &x^2+5x+6=0 \\ &(x+2)(x+3)=0 \\ &x+2=0\;\text{ or }\;x+3=0 \\ &x=-2\;\text{ or }\;x=-3 \\ & \\ &\dfrac{x+4}{x^2+5x+6}\text{ is undefined for }x=-2\text{ or }x=-3. \end{array} $

$try-it.png$ $\PageIndex{1}$

Determine the value for which each rational expression is undefined.

a.$\dfrac{4p}{5q}$ b. $\dfrac{y−1}{3y+2}$ c. $\dfrac{m−5}{m^2+m−6}$

Answer: a. $q=0$
b. $y=−\dfrac{2}{3}$
c. $m=2,m=−3$

Simplify Rational Expressions

A fraction is considered simplified, or reduced, 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.

Definition: SIMPLIFIED RATIONAL EXPRESSION

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

For example,

\[ \begin{array} {l} \dfrac{x+2}{x+3} \text{ is simplified because there are no common factors of } x+2 \text{ and }x+3. \\ \dfrac{2x}{3x} \text{ is not simplified because } x \text{ is a common factor of }2x\text{ and }3x. \\ \end{array} \nonumber\]

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 or expressions where $b\neq 0, c\neq 0,$

\[\text {then } \dfrac{a}{b}=\dfrac{a·c}{b·c} \nonumber\]

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

To simplify rational expressions, we first write the numerator and denominator in factored form. Use the factored form of the denominator to state the variable values that make the expression undefined. 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.

$The rational expression is the quantity 2 times 3 times 7 divided by the quantity 3 times 5 times 7 are 3 and 7. Its common factors are 3 and 7, which are factors of the product. When they are removed, the result is two-fifths. The rational expression is the product of 3 x and the quantity x minus 9 divided by the product of 5 and the quantity x minus 9. The common factor is x minus 9, which is a factor of the product. When it is removed, the result is 3 x divided by 5. The rational expression is the quantity x plus 5 divided by 5. There is an x both the numerator and denomiantor. However, it is a term of the sum in the numerator. The rational expression has no common factors.$

Removing the $x$'s from $\dfrac{x+5}{x}$ would be like cancelling the $2$’s in the fraction $\dfrac{2+5}{2}$, which would imply that $\dfrac{2+5}{2}$ (or $\frac{7}{2}$) equals $5$!

EXAMPLE $\PageIndex{2}$: SIMPLIFY A RATIONAL EXPRESSION

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

Solution:

$Step 1 is to factor the numerator and denominator completely in the rational expression, the quantity x squared plus 5 x plus six divided by the quantity x squared 8 x plus 12. The numerator, x squared plus 5 x plus six, factors into the quantity x plus 2 times the quantity x plus 3. The denominator, x squared 8 x plus 12, factors into the quantity x plus 2 times the quantity x plus 6.$ $Step 2 is to simplify the rational expression, the quantity x plus 2 times the quantity x plus 3 all divided by the quantity x plus 2 times the quantity x plus 6, by dividing out the common factor, x plus 6. The result of removing the common factor is the quantity x plus 3 divided by the quantity x plus 6, where x is not equal to 2 and x is not equal to -6.$

$try-it.png$ $\PageIndex{2}$

Simplify: $\dfrac{x^2−x−2}{x^2−3x+2}$.

Answer: $\dfrac{x+1}{x−1},x\neq 2,x\neq 1$

$try-it.png$ $\PageIndex{3}$

Simplify: $\dfrac{x^2−3x−10}{x^2+x−2}$.

Answer: $\dfrac{x−5}{x−1},x\neq −2,x\neq 1$

We now summarize the steps you should follow to simplify rational expressions.

$how-to.png$ SIMPLIFY A RATIONAL EXPRESSION.

Factor the numerator and denominator completely.
Determine the variable values make the expression undefined.
Simplify by dividing out common factors.

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. Throughout the rest of this section, to let us focus on the work at hand, we will often 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.

EXAMPLE $\PageIndex{3}$

Simplify: $\dfrac{3a^2−12ab+12b^2}{6a^2−24b^2}$.

Solution: $\begin{array} {ll} &\dfrac{3a^2−12ab+12b^2}{6a^2−24b^2} \\ & \\ & \\ \begin{array} {l} \text{Factor the numerator and denominator,} \\ \text{first factoring out the GCF.} \end{array} &\dfrac{3(a^2−4ab+4b^2)}{6(a^2−4b^2)} \\ & \\ &\dfrac{3(a−2b)(a−2b)}{6(a+2b)(a−2b)} \\ & \\ \text{Remove the common factors of }a−2b\text{ and }3. &\dfrac{\cancel{3}(a−2b)\cancel{(a−2b)}}{\cancel{3}·2(a+2b)\cancel{(a−2b)}} \\ &\dfrac{a−2b}{2(a+2b)} \end{array} $

$try-it.png$ $\PageIndex{4}$

Simplify: $\dfrac{2x^2−12xy+18y^2}{3x^2−27y^2}$.

Answer: $\dfrac{2(x−3y)}{3(x+3y)}$

Now we will see how to simplify a rational expression whose numerator and denominator have opposite factors. Section 0.2 introduces opposite notation: the opposite of $a$ is $−a$ and $−a=−1·a$.

For example, the numerical fraction $\frac{7}{−7}$ simplifies to $−1$. Notice that the numerator and denominator are opposites.

The fraction $\frac{a}{−a}$, whose numerator and denominator are opposites also simplifies to $−1$.

\[\begin{array} {ll} \text{Consider this expression } &b−a, (\text{ or } b+(-a)) \\ \text{Rewrite.} &−a+b \\ \text{Factor out }–1. &−1(a−b) \nonumber\end{array} \]

Thus, $b−a$ is the opposite of $a−b$. So the rational expression $\frac{a−b}{b−a}$ simplifies to $−1$.

OPPOSITES IN A RATIONAL EXPRESSION

The opposite of $a−b$ is $b−a$.

\[\dfrac{a−b}{b−a}=−1 \quad a\neq b\nonumber\]

An expression and its opposite divide to $−1$.

We will use this property to simplify rational expressions that contain opposites in their numerators and denominators. Be careful not to treat $a+b$ and $b+a$ as opposites. In addition, order doesn’t matter, so $a+b=b+a$. If $a\neq −b$, then $\frac{a+b}{b+a}=1$.

EXAMPLE $\PageIndex{4}$

Simplify: $\dfrac{x^2−4x−32}{64−x^2}$

Solution

	$.$
Factor the numerator and the denominator.
Recognize the factors that are opposites.	$.$
Simplify.	$.$

$try-it.png$ $\PageIndex{5}$

Simplify: $\dfrac{x^2−4x−5}{25−x^2}$

Answer: $−\dfrac{x+1}{x+5}$

$try-it.png$ $\PageIndex{6}$

Simplify: $\dfrac{x^2+x−2}{1−x^2}$.

Answer: $−\dfrac{x+2}{x+1}$

Multiply Rational Expressions

To multiply rational expressions, we do just what we did with numerical fractions. Multiply the numerators and multiply the denominators. If there are any common factors, we need to remove them to simplify the expression.

MULTIPLICATION OF RATIONAL EXPRESSIONS

If $p$, $q$, $r$, and $s$ are polynomials where $q\neq 0$, $s\neq 0$, then

\[\dfrac{p}{q}·\dfrac{r}{s}=\dfrac{pr}{qs}\nonumber\]

To multiply rational expressions, multiply the numerators and multiply the denominators.

Remember, throughout this chapter, we will assume that all numerical values that would make the denominator be zero are excluded. So in this next example, $x\neq 0$, $x\neq 3$, and $x\neq 4.$ In order to minimize the amount of work, the first step is to factor all polynomials, and then keep them factored.

EXAMPLE $\PageIndex{5}$: MULTIPLY RATIONAL EXPRESSIONS

Multiply: $\dfrac{2x}{x^2−7x+12}·\dfrac{x^2−9}{6x^2}$.

Solution

$Step 1 is to factor each numerator and the denominator completely in 2 x divided by the quantity x squared minus 7 x plus 12 times the rational expression the quantity x squared minus 9 divided by 6 x squared. The denominator, x squared minus 7 x plus 12, factors into the quantity x minus 3 times the quantity x minus 4. The numerator x squared minus 9 factors into the quantity x minus 3 times the quantity x plus 3.$ Step 2 is to multiply the numerators 2 x and the quantity x minus 3 times the quantity x plus 3, and the denominators the quantity x minus 3 times the quantity x minus 4 and 6 x squared. It is helpful to write the monomials in the numerator and in the denominator. first.

$try-it.png$ $\PageIndex{7}$

Multiply: $\dfrac{9x^2}{x^2+11x+30}·\dfrac{x^2−36}{3x^2}$.

Answer: $\dfrac{3(x−6)}{x+5}$

$how-to.png$ MULTIPLY RATIONAL EXPRESSIONS.

Factor each numerator and denominator completely.
Multiply the numerators and denominators.
Simplify by dividing out common factors.

EXAMPLE $\PageIndex{6}$

Multiply: $\dfrac{3a^2−8a−3}{a^2−25}·\dfrac{a^2+10a+25}{3a^2−14a−5}$.

Solution

$\begin{array} {ll} &\dfrac{3a^2−8a−3}{a^2−25}·\dfrac{a^2+10a+25}{3a^2−14a−5} \\ & \\ \begin{array} {ll} \text{Factor the numerators and denominators} \\ \text{and then multiply.} \end{array} &\dfrac{(3a+1)(a−3)(a+5)(a+5)}{(a−5)(a+5)(3a+1)(a−5)} \\ & \\ \begin{array} {l} \text{Simplify by dividing out} \\ \text{common factors.} \end{array} &\dfrac{\cancel{(3a+1)}(a−3)\cancel{(a+5)}(a+5)}{(a−5)\cancel{(a+5)}\cancel{(3a+1)}(a−5)} \\ & \\ \text{Simplify.} &\dfrac{(a−3)(a+5)}{(a−5)(a−5)} \\ & \\ \text{Rewrite }(a−5)(a−5)\text{ using an exponent.} &\dfrac{(a−3)(a+5)}{(a−5)^2} \end{array}$

$try-it.png$ $\PageIndex{8}$

Multiply: $\dfrac{2x^2+5x−12}{x^2−16}·\dfrac{x^2−8x+16}{2x^2−13x+15}$.

Answer: $\dfrac{x−4}{x−5}$

$try-it.png$ $\PageIndex{9}$

Simplify: $\dfrac{4b^2+7b−2}{1−b^2}·\dfrac{b^2−2b+1}{4b^2+15b−4}$.

Answer: $−\dfrac{(b+2)(b−1)}{(1+b)(b+4)}$

Divide Rational Expressions

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

DIVISION OF RATIONAL EXPRESSIONS

If $p$, $q$, $r$, and $s$ are polynomials where $q\neq 0$, $r\neq 0$, $s\neq 0$, then

\[\dfrac{p}{q}÷\dfrac{r}{s}=\dfrac{p}{q}·\dfrac{s}{r}\nonumber\]

Once we rewrite the division as multiplication of the first expression by the reciprocal of the second, we follow the procedure for multiplying rational expressions.

EXAMPLE $\PageIndex{7}$: DIVIDE RATIONAL EXPRESSIONS

Divide: $\dfrac{p^3+q^3}{2p^2+2pq+2q^2}÷\dfrac{p^2−q^2}{6}$.

Solution

$Step 1 is to rewrite the division of the rational expression, the quantity p cubed plus q cubes divided by the quantity 2 p squared plus 2 p q plus 2 q squared divided by the rational expression, the quantity p squared minus q squared all divided by 6. Do this by flipping the rational expression, the quantity p squared minus q squared all divided by 6, and changing division to multiplication. The result is the quantity p cubed plus q cubes divided by the quantity 2 p squared plus 2 p q plus 2 q squared times the quantity 6 divided by the quantity p squared minus q squared.$ $Step 2 is to factor the numerators, the quantity p cubed plus q cubed and 6, and the denominators, the quantity 2 p squared plus 2 p q plus 2 squared and the quantity p squared minus q squared, completely. The result is the quantity p plus q times the quantity p squared minus p q plus q squared all times the quantity 2 times 3 divided by the quantity p minus q times the quantity p plus q.$ Step 3 is to multiply the numerators and denominators. The result is the quantity p plus q times the quantity p squared minus p q plus q squared times 2 times 3 all divided by the 2 times the quantity p squared plus p q plus q squared times the quantity p minus q times the quantity p plus q. $Step 4 is to simplify the expression by dividing out the common factors, the quantity p plus q and 2. The result is 3 times the quantity p squared minus p q plus q squared all divided by the quantity p minus q times the quantity p squared plus p q plus q squared.$

$try-it.png$ $\PageIndex{10}$

Simplify: $\dfrac{x^3−8}{3x^2−6x+12}÷\dfrac{x^2-4}{6}$.

Answer: $\dfrac{2(x^2+2x+4)}{(x+2)(x^2−2x+4)}$

$try-it.png$ $\PageIndex{11}$

Simplify: $\dfrac{2z^2}{z^2−1}÷\dfrac{z^3−z^2+z}{z^3+1}$.

Answer: $\dfrac{2z}{z−1}$

$how-to.png$ DIVISION OF RATIONAL EXPRESSIONS

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

Sometimes, division of rational expressions will be expressed as a complex fraction (fractions within a fraction).

EXAMPLE $\PageIndex{8}$

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

Solution

$\begin{array} {ll} &\dfrac{\dfrac{6x^2−7x+2}{4x−8}}{\dfrac{2x^2−7x+3}{x^2−5x+6}} \\ & \\ \text{Rewrite with a division sign.} &\dfrac{6x^2−7x+2}{4x−8}÷\dfrac{2x^2−7x+3}{x^2−5x+6} \\ & \\ \begin{array} {l} \text{Rewrite as product of first times reciprocal} \\ \text{of second.} \end{array} &\dfrac{6x^2−7x+2}{4x−8}·\dfrac{x^2−5x+6}{2x^2−7x+3} \\ & \\ \begin{array} {l} \text{Factor the numerators and the} \\ \text{denominators, and then multiply.} \end{array} &\dfrac{(2x−1)(3x−2)(x−2)(x−3)}{4(x−2)(2x−1)(x−3)} \\ & \\ \text{Simplify by dividing out common factors.} &\dfrac{\cancel{(2x−1)}(3x−2)\cancel{(x−2)}\cancel{(x−3)}}{4\cancel{(x−2)}\cancel{(2x−1)}\cancel{(x−3)}} \\ \text{Simplify.} &\dfrac{3x−2}{4} \end{array}$

$try-it.png$ $\PageIndex{12}$

Simplify: $\dfrac{\dfrac{3x^2+7x+2}{4x+24}}{\dfrac{3x^2−14x−5}{x^2+x−30}}$.

Answer: $\dfrac{x+2}{4}$

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 $\PageIndex{9}$

Perform the indicated operations: $\dfrac{3x−6}{4x−4}·\dfrac{x^2+2x−3}{x^2−3x−10}÷\dfrac{2x+12}{8x+16}$.

Solution

	$.$
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.	$.$

$try-it.png$ $\PageIndex{13}$

Perform the indicated operations: $\dfrac{4m+4}{3m−15}·\dfrac{m^2−3m−10}{m^2−4m−32}÷\dfrac{12m−36}{6m−48}$.

Answer: $\dfrac{2(m+1)(m+2)}{3(m+4)(m−3)}$

$try-it.png$ $\PageIndex{14}$

Perform the indicated operations: $\dfrac{2n^2+10n}{n−1}÷\dfrac{n^2+10n+24}{n^2+8n−9}·\dfrac{n+4}{8n^2+12n}$.

Answer: $\dfrac{(n+5)(n+9)}{2(n+6)(2n+3)}$

Key Concepts

Determine the values for which a rational expression is undefined.
1. Set the denominator equal to zero.
2. Solve the equation.
Equivalent Fractions Property
If $a$, $b$, and $c$ are numbers where $b\neq 0$, $c\neq 0$, then

$\quad\dfrac{a}{b}=\dfrac{a·c}{b·c}$ and $\dfrac{a·c}{b·c}=\dfrac{a}{b}.$
How to simplify a rational expression.
1. Factor the numerator and denominator completely.
2. Simplify by dividing out common factors.
Opposites in a Rational Expression
The opposite of $a−b$ is $b−a$.

$\quad\dfrac{a−b}{b−a}=−1 \qquad a\neq b$

An expression and its opposite divide to $−1$.
Multiplication of Rational Expressions
If $p$, $q$, $r$, and $s$ are polynomials where $q\neq 0$, $s\neq 0$, then

$\quad\dfrac{p}{q}·\dfrac{r}{s}=\dfrac{pr}{qs}$
How to multiply rational expressions.
1. Factor each numerator and denominator completely.
2. Multiply the numerators and denominators.
3. Simplify by dividing out common factors.
Division of Rational Expressions
If $p$, $q$, $r$, and $s$ are polynomials where $q\neq 0$, $r\neq 0$, $s\neq 0$, then

$\quad\dfrac{p}{q}÷\dfrac{r}{s}=\dfrac{p}{q}·\dfrac{s}{r}$
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.

Glossary

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

simplified rational expression: A simplified rational expression has no common factors, other than $1$, in its numerator and denominator.

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