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Simplify: x 3 + 8 x 2 4 .

x 2 2 x + 4 x 2

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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 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 a . We remember, too, that a = −1 · a .

We simplify the fraction a a , whose numerator and denominator are opposites, in this way:

a a We could rewrite this. 1 · a −1 · a Remove the common factors. 1 −1 Simplify. −1


So, in the same way, we can simplify the fraction x 3 ( x 3 ) :

We could rewrite this. 1 · ( x 3 ) −1 · ( x 3 ) Remove the common factors. 1 −1 Simplify. −1


But the opposite of x 3 could be written differently:

( x 3 ) Distribute. x + 3 Rewrite. 3 x

This means the fraction x 3 3 x simplifies to −1 .

In general, we could write the opposite of a b as b a . So the rational expression a b b a simplifies to −1 .

Opposites in a rational expression

The opposite of a b is b a .

a b b a = −1 a b

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: x 8 8 x .

Solution

x 8 8 x Recognize that x 8 and 8 x are opposites. −1

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Simplify: y 2 2 y .

−1

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Simplify: n 9 9 n .

−1

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

Simplify: 14 2 x x 2 49 .

Solution

.
Factor the numerator and denominator. .
Recognize that 7 x and x 7 are opposites . .
Simplify. .

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Simplify: 10 2 y y 2 25 .

2 y + 5 .

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Simplify: 3 y 27 81 y 2 .

3 9 + y

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Simplify: x 2 4 x 32 64 x 2 .

Solution

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

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Simplify: x 2 4 x 5 25 x 2 .

x + 1 x + 5

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Simplify: x 2 + x 2 1 x 2 .

x + 2 x + 1

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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 .
      a b b a = −1 a 0 , b 0 , a b

Practice makes perfect

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


2 x z
4 p 1 6 p 5
n 3 n 2 + 2 n 8

z = 0 p = 5 6
n = −4 , n = 2

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10 m 11 n
6 y + 13 4 y 9
b 8 b 2 36

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4 x 2 y 3 y
3 x 2 2 x + 1
u 1 u 2 3 u 28

y = 0 x = 1 2
u = −4 , u = 7

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5 p q 2 9 q
7 a 4 3 a + 5
1 x 2 4

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Evaluate Rational Expressions

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

2 x x 1

x = 0
x = 2
x = −1

0 4 1

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4 y 1 5 y 3

y = 0
y = 2
y = −1

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2 p + 3 p 2 + 1

p = 0
p = 1
p = −2

3 5 2 1 5

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x + 3 2 3 x

x = 0
x = 1
x = −2

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y 2 + 5 y + 6 y 2 1

y = 0
y = 2
y = −2

−6 20 3 0

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z 2 + 3 z 10 z 2 1

z = 0
z = 2
z = −2

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a 2 4 a 2 + 5 a + 4

a = 0
a = 1
a = −2

−1 3 10 0

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b 2 + 2 b 2 3 b 4

b = 0
b = 2
b = −2

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x 2 + 3 x y + 2 y 2 2 x 3 y

  1. x = 1 , y = −1
  2. x = 2 , y = 1
  3. x = −1 , y = −2

0 3 4 15 4

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c 2 + c d 2 d 2 c d 3

  1. c = 2 , d = −1
  2. c = 1 , d = −1
  3. c = −1 , d = 2
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m 2 4 n 2 5 m n 3

  1. m = 2 , n = 1
  2. m = −1 , n = −1
  3. m = 3 , n = 2

0 3 5 7 20

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2 s 2 t s 2 9 t 2

  1. s = 4 , t = 1
  2. s = −1 , t = −1
  3. s = 0 , t = 2
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Simplify Rational Expressions

In the following exercises, simplify.

8 m 3 n 12 m n 2

2 m 2 3 n

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3 c 9 5 c 15

3 5

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12 p 240 5 p 100

12 5

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a 2 a 12 a 2 8 a + 16

a + 3 a 4

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x 2 + 4 x 5 x 2 2 x + 1

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y 2 + 3 y 4 y 2 6 y + 5

y + 4 y 5

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v 2 + 8 v + 15 v 2 v 12

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x 2 25 x 2 + 2 x 15

x 5 x 3

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a 2 4 a 2 + 6 a 16

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y 2 2 y 3 y 2 9

y + 1 y + 3

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b 2 + 9 b + 18 b 2 36

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y 3 + y 2 + y + 1 y 2 + 2 y + 1

y 2 + 1 y + 1

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p 3 + 3 p 2 + 4 p + 12 p 2 + p 6

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x 3 2 x 2 25 x + 50 x 2 25

x 2

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q 3 + 3 q 2 4 q 12 q 2 4

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3 a 2 + 15 a 6 a 2 + 6 a 36

a ( a + 5 ) 2 ( a + 3 ) ( a 2 )

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8 b 2 32 b 2 b 2 6 b 80

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−5 c 2 10 c −10 c 2 + 30 c + 100

c 2 ( c 5 )

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4 d 2 24 d 2 d 2 4 d 48

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3 m 2 + 30 m + 75 4 m 2 100

3 ( m + 5 ) 4 ( m 5 )

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5 n 2 + 30 n + 45 2 n 2 18

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5 r 2 + 30 r 35 r 2 49

5 ( r 1 ) r + 7

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3 s 2 + 30 s + 24 3 s 2 48

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t 3 27 t 2 9

t 2 + 3 t + 9 t + 3

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w 3 + 216 w 2 36

w 2 6 w + 36 w 6

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Simplify Rational Expressions with Opposite Factors

In the following exercises, simplify each rational expression.

12 2 x x 2 36

2 x + 6

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4 v 32 64 v 2

4 8 + v

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y 2 11 y + 24 9 y 2

y 8 3 + y

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z 2 9 z + 20 16 z 2

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a 2 5 z 36 81 a 2

a + 4 9 + a

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b 2 + b 42 36 b 2

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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 0.25 x 4206.25 , where x is income. The average tax rate for this income can be found by evaluating the formula 0.25 x 4206.25 x . What would be the average tax rate for a single person earning $50,000?

16.5%

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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 x y 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?

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Writing exercises

Explain how you find the values of x for which the rational expression x 2 x 20 x 2 4 is undefined.

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Explain all the steps you take to simplify the rational expression p 2 + 4 p 21 9 p 2 .

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Self check

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

This figure shows a table with four columns and five rows. The first row is a header row and each column is labeled. The first column header is labeled “I can…”, the second is labeled “Confidently”, the third is labeled “With some help”, and the fourth is labeled “No—I don’t get it!” In the first column under “I can”, the cells read “determine the values for which a rational expression is undefined,” “evaluate rational expressions,” “simplify rational expressions,” and “simplify rational expressions with opposite factors.” The rest of the cells are blank.

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?
Cecilia Reply
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
Heidi Reply
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.
Elbert Reply
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?
Jawad Reply
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.
David Reply
. 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?
jojo Reply
whats the coefficient of 17x
Dwayne Reply
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
A private jet can fly 1,210 miles against a 25 mph headwind in the same amount of time it can fly 1,694 miles with a 25 mph tailwind. Find the speed of the jet
Mikaela Reply
Washing his dad’s car alone, eight-year-old Levi takes 2.5 hours. If his dad helps him, then it takes 1 hour. How long does it take the Levi’s dad to wash the car by himself?
Sam Reply
Ethan and Leo start riding their bikes at the opposite ends of a 65-mile bike path. After Ethan has ridden 1.5 hours and Leo has ridden 2 hours, they meet on the path. Ethan’s speed is 6 miles per hour faster than Leo’s speed. Find the speed of the two bikers.
Mckenzie Reply
Nathan walked on an asphalt pathway for 12 miles. He walked the 12 miles back to his car on a gravel road through the forest. On the asphalt he walked 2 miles per hour faster than on the gravel. The walk on the gravel took one hour longer than the walk on the asphalt. How fast did he walk on the gravel?
Mckenzie
Nancy took a 3 hour drive. She went 50 miles before she got caught in a storm. Then she drove 68 miles at 9 mph less than she had driven when the weather was good. What was her speed driving in the storm?
Reiley Reply
Mr Hernaez runs his car at a regular speed of 50 kph and Mr Ranola at 36 kph. They started at the same place at 5:30 am and took opposite directions. At what time were they 129 km apart?
hamzzi Reply
90 minutes
muhammad
Practice Key Terms 1

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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