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Simplify: $\frac{{x}^{3}+8}{{x}^{2}-4}.$
$\frac{{x}^{2}-2x+4}{x-2}$
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 $\frac{7}{\mathrm{-7}}$ . We know this fraction simplifies to $\mathrm{-1}$ . We also recognize that the numerator and denominator are opposites.
In Foundations , we introduced opposite notation: the opposite of $a$ is $\text{\u2212}a$ . We remember, too, that $\text{\u2212}a=\mathrm{-1}\xb7a$ .
We simplify the fraction $\frac{a}{\text{\u2212}a}$ , whose numerator and denominator are opposites, in this way:
$\begin{array}{cccc}& & & \hfill \phantom{\rule{5em}{0ex}}\frac{a}{\text{\u2212}a}\hfill \\ \text{We could rewrite this.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{1\xb7a}{\mathrm{-1}\xb7a}\hfill \\ \text{Remove the common factors.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{1}{\mathrm{-1}}\hfill \\ \text{Simplify.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\mathrm{-1}\hfill \end{array}$
So, in the same way, we can simplify the fraction
$\frac{x-3}{\text{\u2212}\left(x-3\right)}$ :
$\begin{array}{cccc}\text{We could rewrite this.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{1\xb7\left(x-3\right)}{\mathrm{-1}\xb7\left(x-3\right)}\hfill \\ \text{Remove the common factors.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{1}{\mathrm{-1}}\hfill \\ \text{Simplify.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\mathrm{-1}\hfill \end{array}$
But the opposite of
$x-3$ could be written differently:
$\begin{array}{cccc}& & & \hfill \phantom{\rule{12.5em}{0ex}}\text{\u2212}\left(x-3\right)\hfill \\ \text{Distribute.}\hfill & & & \hfill \phantom{\rule{12.5em}{0ex}}\text{\u2212}x+3\hfill \\ \text{Rewrite.}\hfill & & & \hfill \phantom{\rule{12.5em}{0ex}}3-x\hfill \end{array}$
This means the fraction $\frac{x-3}{3-x}$ simplifies to $\mathrm{-1}$ .
In general, we could write the opposite of $a-b$ as $b-a$ . So the rational expression $\frac{a-b}{b-a}$ simplifies to $\mathrm{-1}$ .
The opposite of $a-b$ is $b-a$ .
An expression and its opposite divide to $\mathrm{-1}$ .
We will use this property to simplify rational expressions that contain opposites in their numerators and denominators.
Simplify: $\frac{x-8}{8-x}.$
$\begin{array}{cccc}& & & \frac{x-8}{8-x}\hfill \\ \text{Recognize that}\phantom{\rule{0.2em}{0ex}}x-8\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}8-x\phantom{\rule{0.2em}{0ex}}\text{are opposites.}\hfill & & & \mathrm{-1}\hfill \end{array}$
Remember, the first step in simplifying a rational expression is to factor the numerator and denominator completely.
Simplify: $\frac{14-2x}{{x}^{2}-49}.$
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: $\frac{{x}^{2}-4x-32}{64-{x}^{2}}.$
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: $\frac{{x}^{2}+x-2}{1-{x}^{2}}.$
$-\frac{x+2}{x+1}$
In the following exercises, determine the values for which the rational expression is undefined.
ⓐ
$\frac{2x}{z}$
ⓑ
$\frac{4p-1}{6p-5}$
ⓒ
$\frac{n-3}{{n}^{2}+2n-8}$
ⓐ
$z=0$
ⓑ
$p=\frac{5}{6}$
ⓒ
$n=\mathrm{-4},n=2$
ⓐ
$\frac{10m}{11n}$
ⓑ
$\frac{6y+13}{4y-9}$
ⓒ
$\frac{b-8}{{b}^{2}-36}$
ⓐ
$\frac{4{x}^{2}y}{3y}$
ⓑ
$\frac{3x-2}{2x+1}$
ⓒ
$\frac{u-1}{{u}^{2}-3u-28}$
ⓐ
$y=0$
ⓑ
$x=-\frac{1}{2}$
ⓒ
$u=\mathrm{-4},u=7$
ⓐ
$\frac{5p{q}^{2}}{9q}$
ⓑ
$\frac{7a-4}{3a+5}$
ⓒ
$\frac{1}{{x}^{2}-4}$
Evaluate Rational Expressions
In the following exercises, evaluate the rational expression for the given values.
$\frac{2x}{x-1}$
ⓐ
$x=0$
ⓑ
$x=2$
ⓒ
$x=\mathrm{-1}$
ⓐ $0$ ⓑ $4$ ⓒ $1$
$\frac{4y-1}{5y-3}$
ⓐ
$y=0$
ⓑ
$y=2$
ⓒ
$y=\mathrm{-1}$
$\frac{2p+3}{{p}^{2}+1}$
ⓐ
$p=0$
ⓑ
$p=1$
ⓒ
$p=\mathrm{-2}$
ⓐ $3$ ⓑ $\frac{5}{2}$ ⓒ $-\frac{1}{5}$
$\frac{x+3}{2-3x}$
ⓐ
$x=0$
ⓑ
$x=1$
ⓒ
$x=\mathrm{-2}$
$\frac{{y}^{2}+5y+6}{{y}^{2}-1}$
ⓐ
$y=0$
ⓑ
$y=2$
ⓒ
$y=\mathrm{-2}$
ⓐ $\mathrm{-6}$ ⓑ $\frac{20}{3}$ ⓒ $0$
$\frac{{z}^{2}+3z-10}{{z}^{2}-1}$
ⓐ
$z=0$
ⓑ
$z=2$
ⓒ
$z=\mathrm{-2}$
$\frac{{a}^{2}-4}{{a}^{2}+5a+4}$
ⓐ
$a=0$
ⓑ
$a=1$
ⓒ
$a=\mathrm{-2}$
ⓐ $\mathrm{-1}$ ⓑ $-\frac{3}{10}$ ⓒ $0$
$\frac{{b}^{2}+2}{{b}^{2}-3b-4}$
ⓐ
$b=0$
ⓑ
$b=2$
ⓒ
$b=\mathrm{-2}$
$\frac{{x}^{2}+3xy+2{y}^{2}}{2{x}^{3}y}$
ⓐ $0$ ⓑ $\frac{3}{4}$ ⓒ $\frac{15}{4}$
$\frac{{c}^{2}+cd-2{d}^{2}}{c{d}^{3}}$
$\frac{{m}^{2}-4{n}^{2}}{5m{n}^{3}}$
ⓐ $0$ ⓑ $-\frac{3}{5}$ ⓒ $-\frac{7}{20}$
$\frac{2{s}^{2}t}{{s}^{2}-9{t}^{2}}$
Simplify Rational Expressions
In the following exercises, simplify.
$-\frac{44}{55}$
$\frac{65}{104}$
$\frac{15xy}{3{x}^{3}{y}^{3}}$
$\frac{36{v}^{3}{w}^{2}}{27v{w}^{3}}$
$\frac{5b+5}{6b+6}$
$\frac{4d+8}{9d+18}$
$\frac{8n-96}{3n-36}$
$\frac{6q+210}{5q+175}$
$\frac{{x}^{2}+4x-5}{{x}^{2}-2x+1}$
$\frac{{v}^{2}+8v+15}{{v}^{2}-v-12}$
$\frac{{a}^{2}-4}{{a}^{2}+6a-16}$
$\frac{{b}^{2}+9b+18}{{b}^{2}-36}$
$\frac{{y}^{3}+{y}^{2}+y+1}{{y}^{2}+2y+1}$
$\frac{{y}^{2}+1}{y+1}$
$\frac{{p}^{3}+3{p}^{2}+4p+12}{{p}^{2}+p-6}$
$\frac{{q}^{3}+3{q}^{2}-4q-12}{{q}^{2}-4}$
$\frac{3{a}^{2}+15a}{6{a}^{2}+6a-36}$
$\frac{a(a+5)}{2(a+3)(a-2)}$
$\frac{8{b}^{2}-32b}{2{b}^{2}-6b-80}$
$\frac{\mathrm{-5}{c}^{2}-10c}{\mathrm{-10}{c}^{2}+30c+100}$
$\frac{c}{2(c-5)}$
$\frac{4{d}^{2}-24d}{2{d}^{2}-4d-48}$
$\frac{3{m}^{2}+30m+75}{4{m}^{2}-100}$
$\frac{3(m+5)}{4(m-5)}$
$\frac{5{n}^{2}+30n+45}{2{n}^{2}-18}$
$\frac{3{s}^{2}+30s+24}{3{s}^{2}-48}$
$\frac{{v}^{3}-1}{{v}^{2}-1}$
$\frac{{w}^{3}+216}{{w}^{2}-36}$
$\frac{{w}^{2}-6w+36}{w-6}$
$\frac{{v}^{3}+125}{{v}^{2}-25}$
Simplify Rational Expressions with Opposite Factors
In the following exercises, simplify each rational expression.
$\frac{b-12}{12-b}$
$\frac{5-d}{d-5}$
$\frac{20-5y}{{y}^{2}-16}$
$\frac{7w-21}{9-{w}^{2}}$
$\frac{{z}^{2}-9z+20}{16-{z}^{2}}$
$\frac{{b}^{2}+b-42}{36-{b}^{2}}$
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?
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 $\frac{{p}^{2}+4p-21}{9-{p}^{2}}.$
ⓐ 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.
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