Unit 8 Rational Functions Homework Packet

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Unit 8 Rational Functions

NAME _____________________________________________ DATE ____________________________ PERIOD _____________

8-1 Study Guide and Intervention

Multiplying and Dividing Rational Expressions

Simplify Rational Expressions A ratio of two polynomial expressions is a rational expression. To simplify a rational
expression, divide both the numerator and the denominator by their greatest common factor (GCF).

Multiplying Rational Expressions For all rational expressions and , . = , if b ≠ 0 and d ≠ 0.
Dividing Rational Expressions


For all rational expressions and , ÷ = , if b ≠ 0, c ≠ 0, and d ≠ 0.



Example: Simplify each expression.

a.
( )

1 11 1 11 1 11

24 5 2 = 2 · 2 · 2 · 3 · · · · · · · = 3
(2 )4 2 · 2 · 2 · 2 · · · · · · · · 2 2

1 1 1 1 11 1 1 1

b. ·


1 11 1 1 11

3 2 3 · 20 2 = 3 · · · · · · 2 ·2 ·5 · · = 2 · 2 · · = 4 2
5 4 9 3 5 · · · · · 3 · 3 · · · · 3 · · · 3 2

1 11 1 11 1

c. + + ÷ + −
– −

2 + 8 + 16 ÷ 2 + 2 − 8 = 2 + 8 + 16 · −1
2 – 2 −1 2 – 2 + 2 −
2 8

11

= ( + 4) ( + 4) ( − 1) = ( + 4)
2( − 1) ( − 2) ( + 4) 2( − 2)

11

Exercises

Simplify each expression.

1. (−2 2)3 2. 4 2 − 1 2 + 9 3. 2+ − 6
20 4 9 − 6 2− 6 – 27

4. 3 3 − 3 · 4 5 5. 2 − 3 · 2 + 4 − 5
6 4 + 1 2 – 25 2 − 4 + 3

6. ( − 3 )2 9 · 3 − 9 7. 6 4 ÷ 18 2
2 − 6 + 2− 9 25 3 5

8. 16 2 − 8 + 1 ÷ 4 2 + 7 − 2 9. 2 − 1 ÷ 4 2 − 1
14 4 7 5 − 3 − 4 + 8
2 10

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8-1 Study Guide and Intervention (continued)

Multiplying and Dividing Rational Expressions

Simplify Complex Fractions A complex fraction is a rational expression with a numerator and/or denominator that is
also a rational expression. To simplify a complex fraction, first rewrite it as a division problem.

−

Example: Simplify .

+ −



3 − 1 = 3 − 1 ÷ 3 2 + 8 − 3 Express as a division problem.
4 Multiply by the reciprocal of the divisor.
Factor and eliminate.
3 2 + 8 − 3 Simplify.

4

= 3 − 1 · 4
+ 8
3 2 − 3

1 3

= (3 − 1) 4 3)
(3 − 1)( +

1

= 3
+ 3

Exercises 2 3 2 − 1

Simplify each expression. 2. 2 2 3. 3 + 2
2 + 1
3 2 4 2 3 2 − − 2

1. 2 2
3 2
2

2 − 100 − 4

4. 2 5. 2+ 6 + 9
3 2 – 31 + 10 2 − 2 − 8

2 3 +

2 − 16 2 2+ 9 + 9

6. + 2 7. + 1
2 + 3 − 4 10 2 + 19 + 6
2 + − 2 5 2 + 7 + 2

+ 2 2 − − 2

8. 2 − 6 + 8 9. 2 + − 6
2 + − 2 + 1
2 − 16
+ 3

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8-2 Study Guide and Intervention

Adding and Subtracting Rational Expressions

LCM of Polynomials To find the least common multiple of two or more polynomials, factor each expression.
The LCM contains each factor the greatest number of times it appears as a factor.

Example 1: Find the LCM of 16 r, 40p , Example 2: Find the LCM of 3 – 3m – 6 and
and 15 . 4 + 12m – 40.

16 2 3r = 24 · 2 · 3 · r 3 2 – 3m – 6 = 3(m + 1)(m – 2)
40p 4 2 = 23 · 5 · p · 4 · 2 4 2 + 12m – 40 = 4(m – 2)(m + 5)
LCM = 12(m + 1)(m – 2)(m + 5)
15 3 4 = 3 · 5 · 3 · 4

LCM = 24 · 3 · 5 · 3 · 4 · 4
= 240 3 4 4

Exercises 2. 8cd 3, 28 2f, 35 4 2
Find the LCM of each set of polynomials.

1. 14a 2, 42b 3, 18 2c

3. 65 4y, 10 2 2, 26 4 4. 11m 5, 18 2 3, 20m 4

5. 15 4b, 50 2 2, 40 8 6. 24 7q, 30 2 2, 45p 3

7. 39 2 2, 52 4c, 12 3 8. 12x 4, 42 2y, 30 2y3

9. 56st 2, 24 2 2, 70 3 3 10. 2 + 3x, 10 2 + 25x – 15

11. 9 2 – 12x + 4, 3 2 + 10x – 8 12. 22 2 + 66x – 220, 4 2 – 16

13. 8 2 – 36x – 20, 2 2 + 2x – 60 14. 5 2 – 125, 5 2 + 24x – 5

15. 3 2 – 18x + 27, 2 3 – 4 2 – 6x 16. 45 2 – 6x – 3, 45 2 – 5

17. 3 + 4 2 – x – 4, 2 + 2x – 3 18. 54 3 – 24x, 12 2 – 26x + 12

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8-2 Study Guide and Intervention (continued)

Adding and Subtracting Rational Expressions

Add and Subtract Rational Expressions To add or subtract rational expressions, follow these steps.

Step 1 Find the least common denominator (LCD). Rewrite each expression with the LCD.
Step 2 Add or subtract the numerators.
Step 3 Combine any like terms in the numerator.
Step 4 Factor if possible.
Step 5 Simplify if possible.

Example: Simplify – .
+ − −

2 2 6 – 12 – 2 4
+ 2 2 −

= 2( + 6 − 2) – ( − 2 + 2) Factor the denominators.
3)( 2)( The LCD is 2(x + 3)(x – 2)(x + 2).
Subtract the numerators.
= 2( + 6( + 2) + 2) – 2( 2 · 2( + 3) + 2) Distribute.
3)( − 2)( + 3)( − 2)( Combine like terms.
Simplify.
= 6( + 2) − 4( + 3)
2( + 3)( − 2)( + 2)

= 6 + 12 − 4 − 12
2( + 3)( − 2)( + 2)

= 2( + 2 2)( + 2)
3)( −

= ( + 3)( 2)( + 2)
−

Exercises

Simplify each expression.

1. −7 + 4 2 2. 2 3 – 1 1
3 2 – –

3. 4 – 15 4. 3 2 + 4 + 5
3 5 + 3 + 6

5. 3 + 3 1 + − 1 6. 4 2 4 + 1 – 5 5
2 + 2 + 2 – 1 − 4 20 2 −

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8-6 Study Guide and Intervention

Solving Rational Equations and Inequalities

Solve Rational Equations A rational equation contains one or more rational expressions. To solve a rational
equation, first multiply each side by the least common denominator of all of the denominators. Be sure to exclude
any solution that would produce a denominator of zero.

Example: Solve + = . Check your solution.
+

9 + 2 1 = 2 Original equation
10 + 5

10(x + 1)(190 + 2 ) = 10(x + 1)(52) Multiply each side by 10(x + 1).
+
1

9(x + 1) + 2(10) = 4(x + 1) Multiply.

9x + 9 + 20 = 4x + 4 Distribute.
5x = –25 Subtract 4x and 29 from each side.
x = –5 Divide each side by 5.

Check 9 + 2 1 = 2 Original equation
10 + 5 x = –5
Simplify.
9 + 2 1 ≟ 2
10 −5 + 5

18 – 10 ≟ 2

20 20 5

2 = 2
5 5

Exercises
Solve each equation. Check your solution.

1. 2 – + 3 = 2 2. 4 − 3 – 4 − 2 = 1 3. 2 + 1 – − 5 = 1
3 6 5 3 3 4 2

4. 3 + 2 + 2 − 1 = 4 5. 4 = + 1 6. + 4 = 10
5 2 – 12 – –
1 2 2

7. NAVIGATION The current in a river is 6 miles per hour. In her motorboat Marissa can travel 12 miles upstream or
16 miles downstream in the same amount of time. What is the speed of her motorboat in still water? Is this a
reasonable answer? Explain.

8. WORK Adam, Bethany, and Carlos own a painting company. To paint a particular house alone, Adam estimates that
it would take him 4 days, Bethany estimates 512 days, and Carlos 6 days. If these estimates are accurate, how long
should it take the three of them to paint the house if they work together? Is this a reasonable answer?

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8-6 Study Guide and Intervention (continued)

Solving Rational Equations and Inequalities

Solve Rational Inequalities To solve a rational inequality, complete the following steps.

Step 1 State the excluded values.
Step 2 Solve the related equation.
Step 3 Use the values from steps 1 and 2 to divide the number line into regions. Test a value in each region to see

which regions satisfy the original inequality.

Example: Solve + = .


Step 1 The value of 0 is excluded since this value would result in a denominator of 0.

Step 2 Solve the related equation.

2 + 4 = 2 Related equation
3 5 3 Multiply each side by 15n.

15n(32 + 54 ) = 15n(23)

10 + 12 = 10n Simplify.

22 = 10n Add.

2.2 = n Divide each side by 10.

Exercises
Solve each inequality. Check your solutions.

1. 3 1 = 3 2. 1 = 4x 3. 1 + 4 = 2
+ 2 5 3

4. 3 – 2 = 1 5. 4 1 + 5 = 2 6. 3 1 + 1 = 2 1
2 4 − 2 − −

7. BASKETBALL Kiana has made 9 of 19 free throws so far this season. Her goal is to make 60% of her free throws. If
Kiana makes her next x free throws in a row, the function f(x) = 9 + 19 + represents Kiana’s new ratio of free throws
made. How many successful free throws in a row will raise Kiana’s percent made to 60%? Is this a reasonable answer?
Explain.

8. OPTICS The lens equation 1 + 1 = 1 relates the distance p of an object from a lens, the distance q of the image of
the object from the lens, and the focal length f of the lens. What is the distance of an object from a lens if the image of the
object is 5 centimeters from the lens and the focal length of the lens is 4 centimeters? Is this a reasonable answer? Explain.

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8-6 Word Problem Practice

Solving Rational Equations and Inequalities

1. HEIGHT Serena can be described as being 8 inches shorter than her sister Malia, or as being 12.5% shorter than

Malia. In other words, 8 8 = 1 , where H is Serena’s height in inches. How tall is Serena?
+ 8

2. CRANES For a wedding, Paula wants to fold 1000 origami cranes.

She does not want to make anyone fold more than 15 cranes. In other words, if N is the number of people enlisted to fold

cranes, Paula wants 1000 ≤ 15. What is the minimum number of people that will satisfy this inequality?


3. RENTAL Carlos and his friends rent a car. They split the $200 rental fee evenly. Carlos, together with just two of his
friends, decide to rent a portable DVD player as well, and split the $30 rental fee for the DVD player evenly among
themselves. Carlos ends up spending $50 for these rentals. Write an equation involving N, the number of friends Carlos
has, using this information. Solve the equation for N.

4. FLIGHT TIME The distance between John F. Kennedy International Airport and Los Angeles International Airport is
about 2500 miles. Let S be
the airspeed of a jet. The wind speed is 100 miles per hour. Because of the wind, it takes longer to fly one way than the
other.

a. Write an equation for S if it takes 2 hours and 5 minutes longer to fly between New York and Los Angeles against
the wind versus flying with the wind.

b. Solve the equation you wrote in part a for S.

c. Write an equation and find how much longer it would take to fly against the wind between New York and Los
Angeles if the wind speed increases to 150 miles per hour and the airspeed of the jet is 525 miles per hour.

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