# 8.1 Simplify rational expressions  (Page 2/5)

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

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

## Simplify rational expressions with opposite factors

Now we will see how to simplify a rational expression whose numerator and denominator have opposite factors. Let’s start with a numerical fraction, say $\frac{7}{-7}$ . We know this fraction simplifies to $-1$ . We also recognize that the numerator and denominator are opposites.

In Foundations , we introduced opposite notation: the opposite of $a$ is $\text{−}a$ . We remember, too, that $\text{−}a=-1·a$ .

We simplify the fraction $\frac{a}{\text{−}a}$ , whose numerator and denominator are opposites, in this way:

$\begin{array}{cccc}& & & \hfill \phantom{\rule{5em}{0ex}}\frac{a}{\text{−}a}\hfill \\ \text{We could rewrite this.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{1·a}{-1·a}\hfill \\ \text{Remove the common factors.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{1}{-1}\hfill \\ \text{Simplify.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}-1\hfill \end{array}$

So, in the same way, we can simplify the fraction $\frac{x-3}{\text{−}\left(x-3\right)}$ :

$\begin{array}{cccc}\text{We could rewrite this.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{1·\left(x-3\right)}{-1·\left(x-3\right)}\hfill \\ \text{Remove the common factors.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{1}{-1}\hfill \\ \text{Simplify.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}-1\hfill \end{array}$

But the opposite of $x-3$ could be written differently:

$\begin{array}{cccc}& & & \hfill \phantom{\rule{12.5em}{0ex}}\text{−}\left(x-3\right)\hfill \\ \text{Distribute.}\hfill & & & \hfill \phantom{\rule{12.5em}{0ex}}\text{−}x+3\hfill \\ \text{Rewrite.}\hfill & & & \hfill \phantom{\rule{12.5em}{0ex}}3-x\hfill \end{array}$

This means the fraction $\frac{x-3}{3-x}$ simplifies to $-1$ .

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

## Opposites in a rational expression

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

$\begin{array}{cccccc}\hfill \frac{a-b}{b-a}=-1\hfill & & & & & \hfill a\ne b\hfill \end{array}$

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.

Simplify: $\frac{x-8}{8-x}.$

## Solution

$\begin{array}{cccc}& & & \frac{x-8}{8-x}\hfill \\ \text{Recognize that}\phantom{\rule{0.2em}{0ex}}x-8\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}8-x\phantom{\rule{0.2em}{0ex}}\text{are opposites.}\hfill & & & -1\hfill \end{array}$

Simplify: $\frac{y-2}{2-y}.$

$-1$

Simplify: $\frac{n-9}{9-n}.$

$-1$

Remember, the first step in simplifying a rational expression is to factor the numerator and denominator completely.

Simplify: $\frac{14-2x}{{x}^{2}-49}.$

## Solution

 Factor the numerator and denominator. Recognize that $7-x\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}x-7\phantom{\rule{0.2em}{0ex}}\text{are opposites}$ . Simplify.

Simplify: $\frac{10-2y}{{y}^{2}-25}.$

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

Simplify: $\frac{3y-27}{81-{y}^{2}}.$

$-\frac{3}{9+y}$

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

## Solution

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

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

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

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

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

## Key concepts

• Determine the Values for Which a Rational Expression is Undefined
1. Set the denominator equal to zero.
2. Solve the equation, if possible.
• Simplified Rational Expression
• A rational expression is considered simplified if there are no common factors in its numerator and denominator.
• 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 $b-a$ .
$\begin{array}{cccccc}\frac{a-b}{b-a}=-1\hfill & & & & & a\ne 0,b\ne 0,\text{a}\ne \text{b}\hfill \end{array}$

## Practice makes perfect

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

$\frac{2x}{z}$
$\frac{4p-1}{6p-5}$
$\frac{n-3}{{n}^{2}+2n-8}$

$z=0$ $p=\frac{5}{6}$
$n=-4,n=2$

$\frac{10m}{11n}$
$\frac{6y+13}{4y-9}$
$\frac{b-8}{{b}^{2}-36}$

$\frac{4{x}^{2}y}{3y}$
$\frac{3x-2}{2x+1}$
$\frac{u-1}{{u}^{2}-3u-28}$

$y=0$ $x=-\frac{1}{2}$
$u=-4,u=7$

$\frac{5p{q}^{2}}{9q}$
$\frac{7a-4}{3a+5}$
$\frac{1}{{x}^{2}-4}$

Evaluate Rational Expressions

In the following exercises, evaluate the rational expression for the given values.

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

$x=0$
$x=2$
$x=-1$

$0$ $4$ $1$

$\frac{4y-1}{5y-3}$

$y=0$
$y=2$
$y=-1$

$\frac{2p+3}{{p}^{2}+1}$

$p=0$
$p=1$
$p=-2$

$3$ $\frac{5}{2}$ $-\frac{1}{5}$

$\frac{x+3}{2-3x}$

$x=0$
$x=1$
$x=-2$

$\frac{{y}^{2}+5y+6}{{y}^{2}-1}$

$y=0$
$y=2$
$y=-2$

$-6$ $\frac{20}{3}$ $0$

$\frac{{z}^{2}+3z-10}{{z}^{2}-1}$

$z=0$
$z=2$
$z=-2$

$\frac{{a}^{2}-4}{{a}^{2}+5a+4}$

$a=0$
$a=1$
$a=-2$

$-1$ $-\frac{3}{10}$ $0$

$\frac{{b}^{2}+2}{{b}^{2}-3b-4}$

$b=0$
$b=2$
$b=-2$

$\frac{{x}^{2}+3xy+2{y}^{2}}{2{x}^{3}y}$

1. $x=1,y=-1$
2. $x=2,y=1$
3. $x=-1,y=-2$

$0$ $\frac{3}{4}$ $\frac{15}{4}$

$\frac{{c}^{2}+cd-2{d}^{2}}{c{d}^{3}}$

1. $c=2,d=-1$
2. $c=1,d=-1$
3. $c=-1,d=2$

$\frac{{m}^{2}-4{n}^{2}}{5m{n}^{3}}$

1. $m=2,n=1$
2. $m=-1,n=-1$
3. $m=3,n=2$

$0$ $-\frac{3}{5}$ $-\frac{7}{20}$

$\frac{2{s}^{2}t}{{s}^{2}-9{t}^{2}}$

1. $s=4,t=1$
2. $s=-1,t=-1$
3. $s=0,t=2$

Simplify Rational Expressions

In the following exercises, simplify.

$-\frac{4}{52}$

$-\frac{1}{13}$

$-\frac{44}{55}$

$\frac{56}{63}$

$\frac{8}{9}$

$\frac{65}{104}$

$\frac{6a{b}^{2}}{12{a}^{2}b}$

$\frac{b}{2a}$

$\frac{15xy}{3{x}^{3}{y}^{3}}$

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

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

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

$\frac{3a+6}{4a+8}$

$\frac{3}{4}$

$\frac{5b+5}{6b+6}$

$\frac{3c-9}{5c-15}$

$\frac{3}{5}$

$\frac{4d+8}{9d+18}$

$\frac{7m+63}{5m+45}$

$\frac{7}{5}$

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

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

$\frac{12}{5}$

$\frac{6q+210}{5q+175}$

$\frac{{a}^{2}-a-12}{{a}^{2}-8a+16}$

$\frac{a+3}{a-4}$

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

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

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

$\frac{{v}^{2}+8v+15}{{v}^{2}-v-12}$

$\frac{{x}^{2}-25}{{x}^{2}+2x-15}$

$\frac{x-5}{x-3}$

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

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

$\frac{y+1}{y+3}$

$\frac{{b}^{2}+9b+18}{{b}^{2}-36}$

$\frac{{y}^{3}+{y}^{2}+y+1}{{y}^{2}+2y+1}$

$\frac{{y}^{2}+1}{y+1}$

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

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

$x-2$

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

$\frac{3{a}^{2}+15a}{6{a}^{2}+6a-36}$

$\frac{a\left(a+5\right)}{2\left(a+3\right)\left(a-2\right)}$

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

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

$\frac{c}{2\left(c-5\right)}$

$\frac{4{d}^{2}-24d}{2{d}^{2}-4d-48}$

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

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

$\frac{5{n}^{2}+30n+45}{2{n}^{2}-18}$

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

$\frac{5\left(r-1\right)}{r+7}$

$\frac{3{s}^{2}+30s+24}{3{s}^{2}-48}$

$\frac{{t}^{3}-27}{{t}^{2}-9}$

$\frac{{t}^{2}+3t+9}{t+3}$

$\frac{{v}^{3}-1}{{v}^{2}-1}$

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

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

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

Simplify Rational Expressions with Opposite Factors

In the following exercises, simplify each rational expression.

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

$-1$

$\frac{b-12}{12-b}$

$\frac{11-c}{c-11}$

$-1$

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

$\frac{12-2x}{{x}^{2}-36}$

$-\frac{2}{x+6}$

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

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

$-\frac{4}{8+v}$

$\frac{7w-21}{9-{w}^{2}}$

$\frac{{y}^{2}-11y+24}{9-{y}^{2}}$

$-\frac{y-8}{3+y}$

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

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

$-\frac{a+4}{9+a}$

$\frac{{b}^{2}+b-42}{36-{b}^{2}}$

## Everyday math

Tax Rates For the tax year 2015, the amount of tax owed by a single person earning between $37,450 and$90,750, can be found by evaluating the formula $0.25x-4206.25,$ where x is income. The average tax rate for this income can be found by evaluating the formula $\frac{0.25x-4206.25}{x}.$ What would be the average tax rate for a single person earning $50,000? 16.5% Work The length of time it takes for two people for perform the same task if they work together can be found by evaluating the formula $\frac{xy}{x+y}.$ If Tom can paint the den in $x=$ 45 minutes and his brother Bobby can paint it in $y=$ 60 minutes, how many minutes will it take them if they work together? ## 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. Explain all the steps you take to simplify the rational expression $\frac{{p}^{2}+4p-21}{9-{p}^{2}}.$ ## 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. #### Questions & Answers Priam has pennies and dimes in a cup holder in his car. The total value of the coins is$4.21 . The number of dimes is three less than four times the number of pennies. How many pennies and how many dimes are in the cup?
Arnold invested $64,000 some at 5.5% interest and the rest at 9% interest how much did he invest at each rate if he received$4500 in interest in one year
List five positive thoughts you can say to yourself that will help youapproachwordproblemswith a positive attitude. You may want to copy them on a sheet of paper and put it in the front of your notebook, where you can read them often.
Avery and Caden have saved $27,000 towards a down payment on a house. They want to keep some of the money in a bank account that pays 2.4% annual interest and the rest in a stock fund that pays 7.2% annual interest. How much should they put into each account so that they earn 6% interest per year? Leika Reply 324.00 Irene 1.2% of 27.000 Irene i did 2.4%-7.2% i got 1.2% Irene I have 6% of 27000 = 1620 so we need to solve 2.4x +7.2y =1620 Catherine I think Catherine is on the right track. Solve for x and y. Scott next bit : x=(1620-7.2y)/2.4 y=(1620-2.4x)/7.2 I think we can then put the expression on the right hand side of the "x=" into the second equation. 2.4x in the second equation can be rewritten as 2.4(rhs of first equation) I write this out tidy and get back to you... Catherine Darrin is hanging 200 feet of Christmas garland on the three sides of fencing that enclose his rectangular front yard. The length is five feet less than five times the width. Find the length and width of the fencing. Ericka Reply Mario invested$475 in $45 and$25 stock shares. The number of $25 shares was five less than three times the number of$45 shares. How many of each type of share did he buy?
let # of $25 shares be (x) and # of$45 shares be (y) we start with $25x +$45y=475, right? we are told the number of $25 shares is 3y-5) so plug in this for x.$25(3y-5)+$45y=$475 75y-125+45y=475 75y+45y=600 120y=600 y=5 so if #$25 shares is (3y-5) plug in y. Joshua will every polynomial have finite number of multiples? cricket Reply a=# of 10's. b=# of 20's; a+b=54; 10a + 20b=$910; a=54 -b; 10(54-b) + 20b=$910; 540-10b+20b=$910; 540+10b=$910; 10b=910-540; 10b=370; b=37; so there are 37 20's and since a+b=54, a+37=54; a=54-37=17; a=17, so 17 10's. So lets check.$740+$170=$910.
. A cashier has 54 bills, all of which are $10 or$20 bills. The total value of the money is \$910. How many of each type of bill does the cashier have?
whats the coefficient of 17x
the solution says it 14 but how i thought it would be 17 im i right or wrong is the exercise wrong
Dwayne
17
Melissa
wow the exercise told me 17x solution is 14x lmao
Dwayne
thank you
Dwayne
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Mckenzie
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90 minutes