Multiplying And Dividing Rational Expression

Rational Expressions and Functions

Multiply and Divide Rational Expressions

Learning Objectives

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

Determine the values for which a rational expression is undefined
Simplify rational expressions
Multiply rational expressions
Split rational expressions
Multiply and divide rational functions

Before yous become started, accept this readiness quiz.

Simplify: $\frac{90y}{15{y}^{2}}.$
If you missed this problem, review (Figure).
Multiply: $\frac{14}{15}·\frac{6}{35}.$
If you missed this problem, review (Figure).
Split up: $\frac{12}{10}÷\frac{8}{25}.$
If you missed this problem, review (Figure).

We previously reviewed the properties of fractions and their operations. Nosotros introduced rational numbers, which are just fractions where the numerators and denominators are integers. In this chapter, we will piece of work with fractions whose numerators and denominators are polynomials. We call this kind of expression a rational expression.

Rational Expression

A rational expression is an expression of the grade $\frac{p}{q},$ where p and q are polynomials and $q\ne 0.$

Here are some examples of rational expressions:

$\begin{array}{cccccccccc}\hfill -\frac{24}{56}\hfill & & & \hfill \phantom{\rule{3em}{0ex}}\frac{5x}{12y}\hfill & & & \hfill \phantom{\rule{3em}{0ex}}\frac{4x+1}{{x}^{2}-9}\hfill & & & \hfill \phantom{\rule{3em}{0ex}}\frac{4{x}^{2}+3x-1}{2x-8}\hfill \end{array}$

Notice that the start rational expression listed above, $-\frac{24}{56}$ , is simply a fraction. Since a constant is a polynomial with degree nix, the ratio of 2 constants is a rational expression, provided the denominator is not aught.

We will do the aforementioned operations with rational expressions that we did with fractions. We will simplify, add, subtract, multiply, divide and use them in applications.

Decide 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, information technology is easy to avoid dividing by nix because we can run across the number in the denominator. In order to avoid dividing by nix in a rational expression, we must not allow values of the variable that will brand the denominator be zero.

And then earlier we brainstorm whatsoever operation with a rational expression, we examine it first to find the values that would make the denominator zero. That fashion, when we solve a rational equation for example, we will know whether the algebraic solutions nosotros find are allowed or not.

Determine the values for which a rational expression is undefined.

Set up the denominator equal to zero.
Solve the equation.

Determine the value for which each rational expression is undefined:

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

The expression volition be undefined when the denominator is zero.

ⓐ

$\begin{array}{cccc}& & & \hfill \phantom{\rule{5em}{0ex}}\frac{8{a}^{2}b}{3c}\hfill \\ \begin{array}{c}\text{Set the denominator equal to zero and solve}\hfill \\ \text{for the variable.}\hfill \end{array}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}3c=0\hfill \\ & & & \hfill \phantom{\rule{5em}{0ex}}c=0\hfill \\ & & & \hfill \phantom{\rule{5em}{0ex}}\frac{8{a}^{2}b}{3c}\phantom{\rule{0.2em}{0ex}}\text{is undefined for}\phantom{\rule{0.2em}{0ex}}c=0.\hfill \end{array}$

ⓑ

$\begin{array}{cccc}& & & \hfill \phantom{\rule{5em}{0ex}}\frac{4b-3}{2b+5}\hfill \\ \begin{array}{c}\text{Set the denominator equal to zero and solve}\hfill \\ \text{for the variable.}\hfill \\ \\ \\ \\ \end{array}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\begin{array}{ccc}\hfill 2b+5& =\hfill & 0\hfill \\ \hfill 2b& =\hfill & -5\hfill \\ \hfill b& =\hfill & -\frac{5}{2}\hfill \end{array}\hfill \\ & & & \hfill \phantom{\rule{5em}{0ex}}\frac{4b-3}{2b+5}\phantom{\rule{0.2em}{0ex}}\text{is undefined for}\phantom{\rule{0.2em}{0ex}}b=-\frac{5}{2}.\hfill \end{array}$

ⓒ

$\begin{array}{cccc}& & & \hfill \frac{x+4}{{x}^{2}+5x+6}\phantom{\rule{1.4em}{0ex}}\\ \begin{array}{c}\text{Set the denominator equal to zero and solve}\hfill \\ \text{for the variable.}\hfill \\ \\ \\ \end{array}\hfill & & & \hfill \begin{array}{c}\hfill {x}^{2}+5x+6=0\phantom{\rule{1em}{0ex}}\\ \hfill \left(x+2\right)\left(x+3\right)=0\phantom{\rule{1em}{0ex}}\\ \hfill x+2=0\phantom{\rule{0.5em}{0ex}}\text{or}\phantom{\rule{0.5em}{0ex}}x+3=0\phantom{\rule{1em}{0ex}}\\ \hfill x=-2\phantom{\rule{0.2em}{0ex}}\text{or}\phantom{\rule{0.2em}{0ex}}x=-3\phantom{\rule{0.2em}{0ex}}\end{array}\\ & & & \hfill \frac{x+4}{{x}^{2}+5x+6}\phantom{\rule{0.2em}{0ex}}\text{is undefined for}\phantom{\rule{0.2em}{0ex}}x=-2\phantom{\rule{0.2em}{0ex}}\text{or}\phantom{\rule{0.2em}{0ex}}x=-3.\end{array}$

Make up one's mind the value for which each rational expression is undefined.

ⓐ $\frac{3{y}^{2}}{8x}$ ⓑ $\frac{8n-5}{3n+1}$ ⓒ $\frac{a+10}{{a}^{2}+4a+3}$

Determine the value for which each rational expression is undefined.

ⓐ $\frac{4p}{5q}$ ⓑ $\frac{y-1}{3y+2}$ ⓒ $\frac{m-5}{{m}^{2}+m-6}$

Simplify Rational Expressions

A fraction is considered simplified if there are no common factors, other than ane, in its numerator and denominator. Similarly, a simplified rational expression has no mutual factors, other than i, in its numerator and denominator.

Simplified Rational Expression

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

For example,

$\begin{array}{c}\frac{x+2}{x+3}\phantom{\rule{0.2em}{0ex}}\text{is simplified because there are no common factors of}\phantom{\rule{0.2em}{0ex}}x+2\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}x+3.\hfill \\ \frac{2x}{3x}\phantom{\rule{0.2em}{0ex}}\text{is not simplified because}\phantom{\rule{0.2em}{0ex}}x\phantom{\rule{0.2em}{0ex}}\text{is a common factor of}\phantom{\rule{0.2em}{0ex}}2x\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}3x.\hfill \end{array}$

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

Equivalent Fractions Holding

If a, b, and c are numbers where $b\ne 0,c\ne 0,$

$\text{then}\phantom{\rule{0.5em}{0ex}}\frac{a}{b}=\frac{a·c}{b·c}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\frac{a·c}{b·c}=\frac{a}{b}.$

Discover that in the Equivalent Fractions Belongings, the values that would brand the denominators zero are specifically disallowed. We see $b\ne 0,c\ne 0$ clearly stated.

To simplify rational expressions, nosotros first write the numerator and denominator in factored grade. Then we remove the common factors using the Equivalent Fractions Holding.

Be very careful as yous remove mutual factors. Factors are multiplied to brand a product. You lot can remove a factor from a product. You cannot remove a term from a sum.

Removing the ten'due south from $\frac{x+5}{x}$ would be like cancelling the ii's in the fraction $\frac{2+5}{2}!$

How to Simplify a Rational Expression

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

Simplify a rational expression.

Gene the numerator and denominator completely.
Simplify by dividing out common factors.

Usually, we leave the simplified rational expression in factored form. This manner, information technology is easy to check that we have removed all the mutual factors.

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

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

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

$\begin{array}{cccc}& & & \hfill \frac{3{a}^{2}-12ab+12{b}^{2}}{6{a}^{2}-24{b}^{2}}\hfill \\ \\ \\ \begin{array}{c}\text{Factor the numerator and denominator,}\hfill \\ \text{first factoring out the GCF.}\hfill \end{array}\hfill & & & \hfill \frac{3\left({a}^{2}-4ab+4{b}^{2}\right)}{6\left({a}^{2}-4{b}^{2}\right)}\hfill \\ \\ \\ & & & \hfill \frac{3\left(a-2b\right)\left(a-2b\right)}{6\left(a+2b\right)\left(a-2b\right)}\hfill \\ \\ \\ \text{Remove the common factors of}\phantom{\rule{0.2em}{0ex}}a-2b\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}3.\hfill & & & \hfill \frac{\overline{)3}\left(a-2b\right)\overline{)\left(a-2b\right)}}{\overline{)3}·2\left(a+2b\right)\overline{)\left(a-2b\right)}}\hfill \\ & & & \hfill \frac{a-2b}{2\left(a+2b\right)}\hfill \end{array}$

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

$\frac{2\left(x-3y\right)}{3\left(x+3y\right)}$

Simplify: $\frac{5{x}^{2}-30xy+25{y}^{2}}{2{x}^{2}-50{y}^{2}}$ .

$\frac{5\left(x-y\right)}{2\left(x+5y\right)}$

Now we will see how to simplify a rational expression whose numerator and denominator have opposite factors. We previously introduced contrary notation: the reverse of a is $\text{−}a$ and $\text{−}a=-1·a.$

The numerical fraction, say $\frac{7}{-7}$ simplifies to $-1$ . We besides recognize that the numerator and denominator are opposites.

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

$\begin{array}{cccc}\text{Let's look at the expression}\phantom{\rule{0.2em}{0ex}}b-a.\hfill & & & \hfill \phantom{\rule{2em}{0ex}}b-a\hfill \\ \text{Rewrite.}\hfill & & & \hfill \phantom{\rule{2em}{0ex}}\text{−}a+b\hfill \\ \text{Factor out}\phantom{\rule{0.2em}{0ex}}-1.\hfill & & & \hfill \phantom{\rule{2em}{0ex}}-1\left(a-b\right)\hfill \end{array}$

This tells usa that $b-a$ is the opposite of $a-b.$

In general, we could write the opposite of $a-b$ as $b-a.$ And then the rational expression $\frac{a-b}{b-a}$ simplifies to $-1.$

Opposites in a Rational Expression

The opposite of $a-b$ is $b-a.$

$\frac{a-b}{b-a}=-1\phantom{\rule{0.5em}{0ex}}a\ne b$

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

We will use this holding to simplify rational expressions that contain opposites in their numerators and denominators. Be conscientious not to care for $a+b$ and $b+a$ as opposites. Recall that in addition, order doesn't affair so $a+b=b+a$ . So if $a\ne \text{−}b$ , so $\frac{a+b}{b+a}=1.$

Simplify: $\frac{{x}^{2}-4x-32}{64-{x}^{2}}.$

Simplify: $\frac{{x}^{2}-4x-5}{25-{x}^{2}}.$

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

Simplify: $\frac{{x}^{2}+x-2}{1-{x}^{2}}.$

$-\frac{x+2}{x+1}$

Multiply Rational Expressions

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

Multiplication of Rational Expressions

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

$\frac{p}{q}·\frac{r}{s}=\frac{pr}{qs}$

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

Remember, throughout this affiliate, we volition assume that all numerical values that would make the denominator be nil are excluded. We will not write the restrictions for each rational expression, but keep in listen that the denominator can never exist zero. And so in this next example, $x\ne 0,$ $x\ne 3,$ and $x\ne 4.$

How to Multiply Rational Expressions

Simplify: $\frac{5x}{{x}^{2}+5x+6}·\frac{{x}^{2}-4}{10x}.$

$\frac{x-2}{2\left(x+3\right)}$

Simplify: $\frac{9{x}^{2}}{{x}^{2}+11x+30}·\frac{{x}^{2}-36}{3{x}^{2}}.$

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

Multiply rational expressions.

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

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

$\begin{array}{cccc}& & & \hfill \phantom{\rule{2em}{0ex}}\frac{3{a}^{2}-8a-3}{{a}^{2}-25}·\frac{{a}^{2}+10a+25}{3{a}^{2}-14a-5}\hfill \\ \\ \\ \begin{array}{c}\text{Factor the numerators and denominators}\hfill \\ \text{and then multiply.}\hfill \end{array}\hfill & & & \hfill \phantom{\rule{2em}{0ex}}\frac{\left(3a+1\right)\left(a-3\right)\left(a+5\right)\left(a+5\right)}{\left(a-5\right)\left(a+5\right)\left(3a+1\right)\left(a-5\right)}\hfill \\ \\ \\ \begin{array}{c}\text{Simplify by dividing out}\hfill \\ \text{common factors.}\hfill \end{array}\hfill & & & \hfill \phantom{\rule{2em}{0ex}}\frac{\overline{)\left(3a+1\right)}\left(a-3\right)\overline{)\left(a+5\right)}\left(a+5\right)}{\left(a-5\right)\overline{)\left(a+5\right)}\overline{)\left(3a+1\right)}\left(a-5\right)}\hfill \\ \\ \\ \text{Simplify.}\hfill & & & \hfill \phantom{\rule{2em}{0ex}}\frac{\left(a-3\right)\left(a+5\right)}{\left(a-5\right)\left(a-5\right)}\hfill \\ \\ \\ \text{Rewrite}\phantom{\rule{0.2em}{0ex}}\left(a-5\right)\left(a-5\right)\phantom{\rule{0.2em}{0ex}}\text{using an exponent.}\hfill & & & \hfill \phantom{\rule{2em}{0ex}}\frac{\left(a-3\right)\left(a+5\right)}{{\left(a-5\right)}^{2}}\hfill \end{array}$

Simplify: $\frac{2{x}^{2}+5x-12}{{x}^{2}-16}·\frac{{x}^{2}-8x+16}{2{x}^{2}-13x+15}.$

$\frac{x-4}{x-5}$

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

$-\frac{\left(b+2\right)\left(b-1\right)}{\left(1+b\right)\left(b+4\right)}$

Divide Rational Expressions

But like nosotros did for numerical fractions, to divide rational expressions, we multiply the first fraction past the reciprocal of the second.

Division of Rational Expressions

If p, q, r, and due south are polynomials where $q\ne 0,r\ne 0,s\ne 0,$ then

$\frac{p}{q}÷\frac{r}{s}=\frac{p}{q}·\frac{s}{r}$

To split rational expressions, 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 and then factor everything and look for common factors.

How to Divide Rational Expressions

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

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

Simplify: $\frac{2{z}^{2}}{{z}^{2}-1}÷\frac{{z}^{3}-{z}^{2}+z}{{z}^{3}+1}.$

$\frac{2z}{z-1}$

Split rational expressions.

Rewrite the division as the product of the starting time rational expression and the reciprocal of the 2nd.
Cistron the numerators and denominators completely.
Multiply the numerators and denominators together.
Simplify by dividing out mutual factors.

Recollect from Apply the Language of Algebra that a complex fraction is a fraction that contains a fraction in the numerator, the denominator or both. Also, recall a fraction bar ways segmentation. A complex fraction is another way of writing division of two fractions.

Split: $\frac{\frac{6{x}^{2}-7x+2}{4x-8}}{\frac{2{x}^{2}-7x+3}{{x}^{2}-5x+6}}.$

$\begin{array}{cccc}& & & \hfill \frac{\frac{6{x}^{2}-7x+2}{4x-8}}{\frac{2{x}^{2}-7x+3}{{x}^{2}-5x+6}}\hfill \\ \\ \\ \text{Rewrite with a division sign.}\hfill & & & \hfill \frac{6{x}^{2}-7x+2}{4x-8}÷\frac{2{x}^{2}-7x+3}{{x}^{2}-5x+6}\hfill \\ \\ \\ \begin{array}{c}\text{Rewrite as product of first times reciprocal}\hfill \\ \text{of second.}\hfill \end{array}\hfill & & & \hfill \frac{6{x}^{2}-7x+2}{4x-8}·\frac{{x}^{2}-5x+6}{2{x}^{2}-7x+3}\hfill \\ \\ \\ \begin{array}{c}\text{Factor the numerators and the}\hfill \\ \text{denominators, and then multiply.}\hfill \end{array}\hfill & & & \hfill \frac{\left(2x-1\right)\left(3x-2\right)\left(x-2\right)\left(x-3\right)}{4\left(x-2\right)\left(2x-1\right)\left(x-3\right)}\hfill \\ \\ \\ \text{Simplify by dividing out common factors.}\hfill & & & \hfill \frac{\overline{)\left(2x-1\right)}\left(3x-2\right)\overline{)\left(x-2\right)}\overline{)\left(x-3\right)}}{4\overline{)\left(x-2\right)}\overline{)\left(2x-1\right)}\overline{)\left(x-3\right)}}\hfill \\ \\ \\ \text{Simplify.}\hfill & & & \hfill \frac{3x-2}{4}\hfill \end{array}$

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

$\frac{x+2}{4}$

Simplify: $\frac{\frac{{y}^{2}-36}{2{y}^{2}+11y-6}}{\frac{2{y}^{2}-2y-60}{8y-4}}.$

$\frac{2}{y+5}$

If we have more than two rational expressions to work with, we notwithstanding follow the same procedure. The first step volition be to rewrite any division every bit multiplication by the reciprocal. Then, nosotros factor and multiply.

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


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

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

$\frac{2\left(m+1\right)\left(m+2\right)}{3\left(m+4\right)\left(m-3\right)}$

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

$\frac{\left(n+5\right)\left(n+9\right)}{2\left(n+6\right)\left(2n+3\right)}$

Multiply and Split Rational Functions

We started this section stating that a rational expression is an expression of the form $\frac{p}{q},$ where p and q are polynomials and $q\ne 0.$ Similarly, we define a rational function as a function of the grade $R\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}$ where $p\left(x\right)$ and $q\left(x\right)$ are polynomial functions and $q\left(x\right)$ is not zero.

Rational Function

A rational function is a part of the form

$R\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}$

where $p\left(x\right)$ and $q\left(x\right)$ are polynomial functions and $q\left(x\right)$ is not zero.

The domain of a rational function is all real numbers except for those values that would crusade division by zero. We must eliminate any values that make $q\left(x\right)=0.$

Determine the domain of a rational office.

Set the denominator equal to naught.
Solve the equation.
The domain is all real numbers excluding the values plant in Step 2.

Find the domain of $R\left(x\right)=\frac{2{x}^{2}-14x}{4{x}^{2}-16x-48}.$

The domain volition be all existent numbers except those values that make the denominator zero. Nosotros will prepare the denominator equal to cypher , solve that equation, and so exclude those values from the domain.

$\begin{array}{cccc}\text{Set the denominator to zero.}\hfill & & & \hfill 4{x}^{2}-16x-48=0\phantom{\rule{0.7em}{0ex}}\\ \text{Factor, first factor out the GCF.}\hfill & & & \hfill 4\left({x}^{2}-4x-12\right)=0\phantom{\rule{0.7em}{0ex}}\\ & & & \hfill 4\left(x-6\right)\left(x+2\right)=0\phantom{\rule{0.7em}{0ex}}\\ \text{Use the Zero Product Property.}\hfill & & & \hfill 4\ne 0\phantom{\rule{1em}{0ex}}x-6=0\phantom{\rule{1em}{0ex}}x+2=0\phantom{\rule{0.7em}{0ex}}\\ \text{Solve.}\hfill & & & \hfill x=6\phantom{\rule{1em}{0ex}}x=-2\\ & & & \begin{array}{c}\text{The domain of}\phantom{\rule{0.2em}{0ex}}R\left(x\right)\phantom{\rule{0.2em}{0ex}}\text{is all real numbers}\hfill \\ \text{where}\phantom{\rule{0.2em}{0ex}}x\ne 6\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}x\ne \text{−}2.\hfill \end{array}\hfill \end{array}$

Discover the domain of $R\left(x\right)=\frac{2{x}^{2}-10x}{4{x}^{2}-16x-20}.$

The domain of $R\left(x\right)$ is all real numbers where $x\ne 5$ and $x\ne \text{−}1.$

Find the domain of $R\left(x\right)=\frac{4{x}^{2}-16x}{8{x}^{2}-16x-64}.$

The domain of $R\left(x\right)$ is all real numbers where $x\ne 4$ and $x\ne \text{−}2.$

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 split rational functions, we divide the resulting rational expressions on the correct side of the equation using the aforementioned techniques we used to dissever rational expressions.

Primal Concepts

Make up one's mind the values for which a rational expression is undefined.
1. Set the denominator equal to goose egg.
2. Solve the equation.
Equivalent Fractions Holding
If a, b, and c are numbers where $b\ne 0,c\ne 0,$ then $\frac{a}{b}=\frac{a·c}{b·c}$ and $\frac{a·c}{b·c}=\frac{a}{b}.$
How to simplify a rational expression.
1. Gene the numerator and denominator completely.
2. Simplify by dividing out common factors.
Opposites in a Rational Expression
The contrary of $a-b$ is $b-a.$

$\frac{a-b}{b-a}=-1\phantom{\rule{8em}{0ex}}a\ne b$

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

$\phantom{\rule{8em}{0ex}}\frac{p}{q}·\frac{r}{s}=\frac{pr}{qs}$
How to multiply rational expressions.
1. Gene each numerator and denominator completely.
2. Multiply the numerators and denominators.
3. Simplify by dividing out common factors.
Sectionalization of Rational Expressions
If p, q, r, and s are polynomials where $q\ne 0,r\ne 0,s\ne 0,$ and so

$\phantom{\rule{8em}{0ex}}\frac{p}{q}÷\frac{r}{s}=\frac{p}{q}·\frac{s}{r}$
How to divide rational expressions.
1. Rewrite the partitioning as the production of the first rational expression and the reciprocal of the second.
2. Cistron the numerators and denominators completely.
3. Multiply the numerators and denominators together.
4. Simplify by dividing out common factors.
How to decide the domain of a rational function.
1. Set the denominator equal to zip.
2. Solve the equation.
3. The domain is all real numbers excluding the values found in Step 2.

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.

Simplify Rational Expressions

In the following exercises, simplify each rational expression.

$-\frac{44}{55}$

$-\frac{4}{5}$

$\frac{56}{63}$

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

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

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

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

$\frac{8}{3}$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$-1$

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

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

$-\frac{5}{y+4}$

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

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

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

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

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

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

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

Multiply Rational Expressions

In the post-obit exercises, multiply the rational expressions.

$\frac{12}{16}·\frac{4}{10}$

$\frac{3}{10}$

$\frac{32}{5}·\frac{16}{24}$

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

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

$\frac{12{a}^{3}b}{{b}^{2}}·\frac{2a{b}^{2}}{9{b}^{3}}$

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

$\frac{p\left(p-4\right)}{2\left(p-9\right)}$

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

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

$\frac{y-5}{3\left(y+5\right)}$

$\frac{{z}^{2}+3z}{{z}^{2}-3z-4}·\frac{z-4}{{z}^{2}}$

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

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

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

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

$\frac{\left(c+2\right)\left(c+2\right)}{\left(c-2\right)\left(c-3\right)}$

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

$\frac{2{m}^{2}-3m-2}{2{m}^{2}+7m+3}·\frac{3{m}^{2}-14m+15}{3{m}^{2}+17m-20}$

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

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

Divide Rational Expressions

In the following exercises, dissever the rational expressions.

$\frac{v-5}{11-v}÷\frac{{v}^{2}-25}{v-11}$

$-\frac{1}{v+5}$

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

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

$\frac{3s}{s+4}$

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

$\frac{{p}^{3}+{q}^{3}}{3{p}^{2}+3pq+3{q}^{2}}÷\frac{{p}^{2}-{q}^{2}}{12}$

$\frac{4\left({p}^{2}-pq+{q}^{2}\right)}{\left(p-q\right)\left({p}^{2}+pq+{q}^{2}\right)}$

$\frac{{v}^{3}-8{w}^{3}}{2{v}^{2}+4vw+8{w}^{2}}÷\frac{{v}^{2}-4{w}^{2}}{4}$

$\frac{{x}^{2}+3x-10}{4x}÷\left(2{x}^{2}+20x+50\right)$

$\frac{x-2}{8x}$

$\frac{2{y}^{2}-10yz-48{z}^{2}}{2y-1}÷\left(4{y}^{2}-32yz\right)$

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

$\frac{2a-7}{5}$

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

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

$3\left(3c-5\right)$

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

For the following exercises, perform the indicated operations.

$\frac{10{m}^{2}+80m}{3m-9}·\frac{{m}^{2}+4m-21}{{m}^{2}-9m+20}÷\frac{5{m}^{2}+10m}{2m-10}$

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

$\frac{4{n}^{2}+32n}{3n+2}·\frac{3{n}^{2}-n-2}{{n}^{2}+n-30}÷\frac{108{n}^{2}-24n}{n+6}$

$\frac{12{p}^{2}+3p}{p+3}÷\frac{{p}^{2}+2p-63}{{p}^{2}-p-12}·\frac{p-7}{9{p}^{3}-9{p}^{2}}$

$\frac{\left(4p+1\right)\left(p-4\right)}{3p\left(p+9\right)\left(p-1\right)}$

$\frac{6q+3}{9{q}^{2}-9q}÷\frac{{q}^{2}+14q+33}{{q}^{2}+4q-5}·\frac{4{q}^{2}+12q}{12q+6}$

Multiply and Separate Rational Functions

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

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

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

For the following exercises, observe $R\left(x\right)=f\left(x\right)·g\left(x\right)$ where $f\left(x\right)$ and $g\left(x\right)$ are given.

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

$\phantom{\rule{1.4em}{0ex}}g\left(x\right)=\frac{{x}^{2}-64}{{x}^{2}-8x}$

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

$\phantom{\rule{1.4em}{0ex}}g\left(x\right)=\frac{x-5}{{x}^{2}}$

For the following exercises, find $R\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}$ where $f\left(x\right)$ and $g\left(x\right)$ are given.

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

$\phantom{\rule{1.4em}{0ex}}g\left(x\right)=\frac{4{x}^{3}+28{x}^{2}}{{x}^{2}+11x+28}$

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

$\phantom{\rule{1.4em}{0ex}}g\left(x\right)=\frac{12{x}^{2}+36x}{{x}^{2}-11x+18}$

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 lot accept to simplify the rational expression $\frac{{p}^{2}+4p-21}{9-{p}^{2}}.$

ⓐ Multiply $\frac{7}{4}·\frac{9}{10}$ and explain all your steps.

ⓑ Multiply $\frac{n}{n-3}·\frac{9}{n+3}$ and explicate all your steps.

Answers will vary.

ⓐ Divide $\frac{24}{5}÷6$ and explain all your steps.

ⓑ Divide $\frac{{x}^{2}-1}{x}÷\left(x+1\right)$ and explicate all your steps.

Self Check

ⓐ Afterwards completing the exercises, employ this checklist to evaluate your mastery of the objectives of this section.

This table has four columns and six rows. The first row is a header and it labels each column,

ⓑ If most of your checks were:

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

…with some aid. This must be addressed quickly as topics yous do non master become potholes in your road to success. Math is sequential – every topic builds upon previous work. Information technology is important to make sure you have a strong foundation earlier you movement on. Who tin you ask for help? Your fellow classmates and instructor are good resource. 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 y'all must not ignore it. You demand to get help immediately or you lot volition quickly be overwhelmed. Run across your teacher as shortly as possible to talk over your situation. Together you tin come up with a plan to become you lot the help you need.