What You've learned
California Content Standards
10.0 Add, subtract, and multiply monomials and polynomials. Solve multi-step
problems, including word problems, by using these techniques.
11.0 Apply basic factoring techniques to second- and simple third-degree
polynomials.
14.0 Solve a quadratic equation by factoring.
15.0 Apply algebraic techniques to solve percent mixture problems.
I•
Adding and Subtracting Fractions
for Help to the Lesson in green.
(Skills Handbook page 600)
Add or subtract. Write each answer in simplest form.
1· 32 + 21
3
6
2· 13
+ 13
Solving Quadratic Equations
16
3
3· 25
+ 10
4.
~
-16
(Lesson 9-S)
Solve by factoring.
5. c2 + 4c - 32 = 0
6. m 2 - 9m + 14 = 0
8. s2
9. h 2
-
4s = 12
Simplifying Expressions
-
6h = 27
7. p 2 + 6p + 5 = 0
10. k 2 = 4k
(Lesson 7-5)
Simplify each expression.
11.
12. 81r105 6
6w3x2
2wx
(3r2s)4
Solving Radical Equations
(Lesson 1 0-4)
Solve each radical equation. If there is no solution, write no solution.
14.
Vx-
4 = 6
15.
Finding the Domain and Range
v'3X + 5 = 2
16. 2x = V3x + 1
(Lesson 10-5)
Find the domain and range of each function.
17. f(x) = 5-
528
Vx
Cha pte r 11
18. y = -2 + v'3X
19. y = v 10 - 3x
.A. When you pluck a guitar
string, it vibrates according
to the rational equation
What You'll Learn Next
f,
California Content Standards
10.0 Divide monomials and polynomials. Solve multi-step problems, including
word problems, by using these techniques.
12.0 Simplify fractions with polynomials in the numerator and denominator by
factoring both and reducing them to the lowest terms.
13.0 Add, subtract, multiply, and divide rational expressions. Solve both
computationally and conceptually challenging problems by using these
techniques.
f = where f is the
frequency of string vibration
(cycles/s}, k is a constant, and
L is the length of the string
(em). In Lesson 11-5, you will
solve real -world problems
involving rational equations.
15.0 Apply algebraic techniques to solve rate problems and work problems.
·>~ English and Spanish Audio Online
• rational equation (p. 550)
• rational expression (p. 530)
Simplifying Rational
Expressions
California Content Standards
12.0 Simplify fractions with polynomials in the numerator and denominator by factoring both
and reducing them to the lowest terms. Introduce, Develop, Master
for Help
@ Check Skills You'll Need
What You'll Learn
• To simplify rational
expressions
Skills Handbook page 598
and Lesson 8-5
Write each fraction in simplest form.
1· 28
... And Why
3· 25
35
15
2· -24
Factor each quadratic expression.
To find the baking time for
bread, as in Example 4
4. x 2 + x - 12
5. x 2 + 6x + 8
6. x 2 - 2x- 15
7. x 2
8. x 2
9. x 2 - 7x + 12
+
8x
+ 16
..,~ New Vocabulary •
-
x - 12
rational expression
r-simplrlylng Rational Expressions
l
Fractions like ~, 2 , and ~ are rational numbers. An expression that can be written
. h f
polynomial .
.
I
.
m t e orm polynomial 1s a rat1ona express1on. Here are some examples:
x+2
3
1
X-
X
x
2
x2 - 5
- lOx + 25
Of course, the value of the expression in the denominator cannot be zero, since
division by zero is undefined. For the rest of this chapter, assume that the values of
the variables that make the denominator zero are excluded from the domain.
Like rational numbers, a rational expression is in simplest form if the numerator
and denominator have no common factors except 1. For example, z l~z is in
simplest form since no factor of 10z is a factor of z + 5.
5
Simplifying a Rational Expression
Simplify 6x + 12
X+ 2
6x + 12 _ 6(x + 2)
x + 2 - x+2
Factor the numerator. The denominator cannot be factored.
6.(.x-+--21 1
Divide out the common factor x
l.x.-+-2
Q
=6
+ 2.
Simplify.
@ CA Standards Check (!) Simplify each expression.
15b
a. 25b2
12c2
b. 3c + 6
4m-2
c. 2m- 1
d 20 + 4t
• t +5
Recall that you learned to factor quadratic expressions in Lessons 8-5 and 8-6.
You may need to factor a quadratic expression to simplify a rational expression.
530
Chapter 11
Rational Expressions and Equations
Simplifying a Rational Expression
Simplify
x
2
2x - 12 .
- 7x + 6
2x - 12 _
2(x - 6)
x 2 - 7x + 6 - (x - 6)(x - 1)
Factor the numerator and the denominator.
2~1
= 1 ~ (x - 1)
Divide out the common factor x - 6.
_ _ 2_
Simplify.
-x - 1
@CA Standards Check
2 } Simplify each expression.
a • 23x
X
+ 12
-
20
X -
c
8a + 16
• 2a 2 + Sa + 2
z,_-----=2=---b. ----=-'2=
2
z - 4z + 3
The numerator and denominator of
2 - c - 6
d. --'c:2:;:--=-------=-
c
+ 5c + 6
3=~ are opposites. To simplify the expression,
you can factor - 1 from 3 - x to get - 1(- 3 + x ), which you can rewrite as
-1(x - 3). Then simplify ;~ - ~
3
_, , .
Recognizing Opposite Factors
Simplify Sx - 15
.
2
9- x
5x - 15
5(x - 3)
9 - x2 = (3 - x) (3 + x)
Factor the numerator and the denominator.
_
5(x - 3)
- - 1(x - 3)(3 + x)
-
Factor -1 from 3 - x.
S{.x---311
- - 11{x---3J(x + 3)
5
Divide out the common factor x
- 3.
Simplify.
- X+ 3
@CA Standards Check ® Simplify each expression.
x-4
-x
8-m
b. - 2 m -64
3 ·4
c.
8- 4r
2
r + 2r - 8
d. 2c2 - 2
3- 3c2
You can use a rational expression to model some real-world situations.
Evaluating a Rational Expression
The baking time for bread depends, in part, on its size and shape. A good
approximation for the baking time, in minutes, of a cylindrical loaf is
~~rfa~~~~~~, or /<;~,where the radius r and the length h of the baked loaf are in
inches. Find the baking time for a loaf that is 5 inches long and has a radius of
4 inches. Round your answer to the nearest minute.
30rh _ 30( 4)(5)
r+h - 4 + 5
For a given volume of dough,
the greater the surface area is,
the shorter the baking time.
Substitute 4 for rand 5 for h.
-----g
600
Simplify.
= 67
Round to the nearest whole number.
The baking time is approximately 67 minutes.
Lesson 11-1
Simplifying Rational Expressions
531
{i! CA Standards Check ~ a. Find the baking time for a loaf that is 4 inches long and has a radius of 3 inches.
Round your answer to the nearest minute.
. 60
. volume
.
b. Cn"t"1ca I Th"m k"mg Th e ratiO
surface
area f or a cy1.m der IS
60nrzh • s·Imp l"f
I y th.IS
2nr 2 + 2nrh
expression to show that it is the same as the expression evaluated in Example 4.
For more exercises, see Extra Skills and Word Problem Practice.
•~ » zu_cu:a:w:
0
Practice by Example
Example 1
for
Help
Simplify each expression.
1 6a + 9
•
(page 530)
12
2p- 24
4· 4p- 48
Example 2
(page 531)
2x 2
2
+ 2x
• 3x + 3x
7
10. w~ + 7w
w - 49
2
13. c 2 - 6c + 8
c + c - 6
Example 3
(page 531)
2m- 5
3· 6m
- 15
2
5• 3xX -- 39x
6 3x +2 6
• 28x 4
•
8 2b- 8
• b2
9.
16
-
2 + 2a + 1
11• a Sa
+5
14 b2 + 8b + 15
•
b + 5
5 - 4n
16• 4n5
17. 212 - 4t
m- 2
19· 42m
20. v - 5 2
(page 531)
Example 4
2 4x3
t -2t-3
m2
m + 6
- m- 42
2
12. m 2 + 7 m + 12
m + 6m + 8
m + 4
15.
m 2 +2m- 8
- 8
18· 4m
4- 2m
21.
25- v
3x
4-w
w2 - 8w + 16
Use the expression/~\ to estimate the baking time in minutes for each type of
bread. Round your answer to the nearest minute.
23. pita: r = 3.5 in., h = 0.5 in.
22. baguette: r = 1.25 in. , h = 26 in.
24. biscuit: r = 1 in. , h = 0.75 in.
0
Apply Your Skills
Simplify each expression.
2r 2 + 9r - 5
25. -=;2;......---'-----"--'--------"'-r + lOr + 25
26 . 7z~ + 23z + 6
z + 2z - 3
27. 5t 2 + 6t - 8
3t 2 + St - 2
28.
2
29 . 3z +. 12z
30.
32a3
16a 2 - 8a
2
31. 4a
- 8a - 5
~
15 - a - 2a
532
Chapter 11
Rational Expressions and Equations
z
i
16m + 3m2
m -3m- 28
32. 16
2s2
s
i
s
33. 10c2+ c2 - 3c3
Sc - 6c - 8
34. a. To keep heating costs down for a structure, architects want the ratio of
surface area to volume as small as possible. Find an expression for the ratio
of the surface area to volume for each shape.
i. square prism
ii. cylinder
uh
4=71h
~
b
b. Find the ratio for each figure when b = 12 ft , h = 18 ft , and r = 6 ft.
A cylinder with a height of
500ft and a volume of
1 X 10 6 ft 3 has about 89% of
the surface area of a square
prism with the same height
and volume.
35. Write an expression that has 2 and -3 excluded from the domain.
Error Analysis Explain the error the student made in simplifying each rational
expression. Then simplify the expression correctly.
37.
36.
=
38. Writing Explain why
1is not the same as x -
2
~
;
3.
. d h
. area of shaded part ~
h fi
s· l"f
.
Fm
t e ratio area of whole figure .or eac gore. Imp 1 y your expressiOn.
40.
39.
Homework Video Tutor
3x
I
I
1Sw
Visit: PHSchool.com
Web Code: bae-11 01
~'----y---J~
Sw
6
6
2t
42.
4L 3y {
/~r7
}8
'-------y-----J'-----y-J
8
2y
Challenge
t+6
'----y---J~
3t
6
Simplify each expression.
43.
m 2- n 2
2
m + llmn + lOn
-
44. a~ - 5ab + 6b2
a
+
2ab - 8b 2
45.
36v 2 - 49w2
2
18v + 9vw - 14w
-
Math Reasoning Determine whether each statement is sometimes, always, or
never true for real numbers a and b.
46.
i
2
=
2
47. ab} = ab
nline Lesson Quiz Visit: PHSchool.com, Web Code: baa-1 101
b
48 a 2 + 6a - 5 = a + 5
•
2a + 2
2
533
For California Standards Tutorials, visit PHSchool.com. Web Code: baq-9045
Alg112.0
49. Which expression is in simplest form?
®
Alg112.0
_L_±__1_
t2 - 1
®
2n -1
©
n2+4
c-7
7-c
2r- 4
®8+ 6r
50. What is the ratio of the area of the small circle to
the area of the large circle?
®~
9n
([) 49x
Alg117.0
3
©
7n
®
49
9
51. Which values of x are NOT in the domain of
(x + l)(x - 3)?
(x - 2)(x - 5) ·
®
-1 and 3
([) 2and5
Alg1 21.0
Alg1 5.0
©
1 and -3
®
-2and - 5
52. Which equation represents the graph of y = x 2
+ 6 shifted 3 units up?
®
y=x2 +3
©
y=3x 2 +6
®
y=x2 +9
®
y=3x2 +9
53. Water is leaking from a 350-mL container at a rate of 2.5 mL per hour. The
equation w = -2.5h + 350 gives the number of milliliters of water win the
container after h hours. How long will the container take to empty?
Alg1 24.0
®
140 hours
©
210 hours
®
350hours
®
490hours
54. Look at the equations shown below.
31 = 3
32 = 9
33 = 27
35 = 243
34 = 81
36 = 729
Use the pattern in the equations to find the value of the ones digit in 399 .
®
Lesson 10-1
for
Help
534
Chapter 11
®
3
©
7
®
9
Simplify each radical expression.
\180
55.
V20 · VIO
56.~
57.
58.
(2;;;
-yzs;;;s
59.
V9Qhfk4
60.
v'6. V8
62.
v'9X. vTIX
63.
f28Y5
-vw
-y-
61.
Lesson 9-1
1
!72
-v 2:!-
VIO
Order each group of quadratic functions from widest to narrowest graph.
~x 2
64. y = x 2 , y = 3x 2 ,y = -2x 2
65. y = ix 2 ,y = tx 2 ,y =
66. y = 2x 2 ,y = 0.5x 2 ,y = -4x 2
67. y = -x 2 ,y = 2.3x 2 ,y = -3.8x 2
Rational Expressions and Equations
Multiplying and Dividing
Rational Expressions
~ California Content Standards
2.0 Understand and use such operations as finding the reciprocal. Master
13.0 Multiply and divide rational expressions. Solve both computationally and conceptually
challenging problems by using these techniques. Introduce
What You'll Learn
• To multiply rational
expressions
Lessons 7-3 and 8-6
Simplify each expression.
1. ,2 .
• To divide rational expressions
,s
4. 3x 4 • 2x 5
... And Why
To find loan payments,
as in Exercises 37- 39
(ru) for Help
@ Check Skills You'll Need
Factor each polynomial.
7. 2c 2 + 15c + 7
c2
2. b 3 • b 4
3. c7
5. 5n 2 • n 2
6. 15a\-3a2)
8. 15t 2 - 26t + 11
9. 2q 2 + 11q + 5
-7-
Multiplying Rat1onal Expressions
Multiplying rational expressions is similar to multiplying rational numbers.
If a, b, c, and d represent polynomials (with b =!= 0 and d =!= 0), then Jj • J = gd.
Multiplying Rational Expressions
Multiply.
a .3. . A_
x2
•x
.3. .
x
b.
Multiply the numerators and multiply the
denominators.
A_- 12
x2
-
X
X+
x3
X-
4 •
X-
3
2
x
x - 3
x(x - 3)
x + 4 · x - 2 = (x + 4)(x - 2)
6
a
@ CA Standards Check CV Multiply.
Multiply the numerators and multiply the
denominators. leave the answer in factored form.
x-5 x- 7
b. x+3 · - x -
-2
a
a. 2 " 3
As with rational numbers, the product gd may not be in simplest form. Look for
factors common to the numerator and the denominator to divide out.
Using Factoring
Multiply 2 x + 1 and _______fu
2
2x + 1
3
@ CA Standards Check
3
4x - 1
6x
2x + 1
6x
4x~ = - 3 - . (2x + 1)(2x- 1)
Factor the denominator.
_ Zx---F"ll .
62x
12'
1~(2x - 1)
Divide out the common factors.
_
2x
- 2x - 1
Simplify.
2) Multiply x - 2 and -8x - 16
8x
x2
-
4 ·
lesson 11-2
Multiplying and Dividing Rational Expressions
535
You can also multiply a rational expression by a polynomial.
Multiplying a Rational Expression by a Polynomial
Multiply ~~
! ~ and s
3s + 2 . ( 2
2s + 4
s
+ 5 + 6) = 3s + 2 . (s + 2)(s + 3)
s
2(s + 2)
1
2
·
@CA Standards Check
+ 5s + 6.
3s + 2
_
-2 1~ ·
Factor.
~ 1 (s + 3)
1
Divide out the common factor
s + 2.
_ (3s + 2) (s + 3)
•
-
Q) Multiply.
a.~ · (c 3
-
Leave in factored form .
2
b. v ~ 3 · (v 2 - 2v - 15)
c)
c. (m - 1) . 4m + 8
m2- 1
r-oividing Rational Expressions
Recall that~ -:- ~ = ~ · ~,where b
=!=
0, c
=!=
0, and d
=!=
0.
When you divide rational expressions that can be factored, first rewrite the
expression using the reciprocal before dividing out common factors.
Dividing Rational Expressions
2
Divide a + ?a + 10 b
a - 6
____q____±_2
y a2 - 36.
2
Multiply by a - 36
a+ 5'
a 2 + ?a + 10 -;-. a2 + 5 = a2 + ?a + 10 . a2 - 36
a - 6
a - 36
a - 6
a + 5
The vinculum. or fraction bar,
is a grouping symbol.
a+ 5
the reciprocal of a 2
(a + 2)(a + 5) (a - 6)(a + 6)
(a - 6)
·
a + 5
1
~ (a
2){a-+-5}1
@CA Standards Check
(a +
=
1~
= (a + 2)(a + 6)
•
•
-
36 .
Factor.
+ 6)
Divide out the common factors.
1a-+-5
Leave in factored form .
f41 Divide.
a.
a-2_:_a-2
ab
·
a
2
2n-3
c. 6n 2 - 5n - 6 -;-. 2n - n - 3
n + 1
b 5m + 10 _:_ 7m + 14
• 2m - 20 · 14m - 20
The reciprocal of a polynomial such as 5x 2 + 5x is~
Sx
+ Sx
Dividing a Rational Expression by a Polynomial
2
Divide x + }~ + 2 by (5x 2
+ 5x ).
x2 + 3x + 2 . 5x2 + 5x _ x 2 + 3x + 2
1
4x
-;1
4x
. 5x 2 + 5x
(x + l)(x + 2)
=
4x
{x-+-!)1(x + 2)
4x
•
536
Chapter 11
Rational Expressions and Equations
_x+2
- 20x 2
1
· 5x( x + 1)
1
· 5x1{x-+-t)
Multiply by the reciprocal
of Sx 2
+ Sx.
Factor.
Divide out the common
factor.
Simplify.
@CA Standards Check
\2) D ivide.
3
a. 3 ~
.,
7
(
..
,I
b y + 3
-15x 5)
•y + 2
2
Z
z2
c.
(y + 2)
7
+ 2z - 15 .
+ 9z + 20 -;- ( z - 3)
For more exercises, see Extra Skills and Word Problem Practice.
•
.
-------·-----·----·-·-
Practice by Example
Multiply.
Example 1
1. 3 . 12
2. ?.
m- 2
m
4• m + 2 · m- 1
c + 1
7• 2c 4c
+ 2 · c - 1
2x
x - 1
5" x+l
· -3-
2
2
6 · 56x " x+l
8 5x 3
9 _]L_ . 3t- 6
for
Help
(page 535)
Example 2
(page 535)
Example 3
7
5x
• x2
~t! 34 • (t 2
t - 6)
-
3 _i_ • _8_
• 3a 2 a3
• 3x4
~~ ~ ~ · (9m 2
14.
t2
• t - 2
6x
x - 5
6x + 9
11• 4x
+ 6 · 3x - 15
m- 2 2m+ 6
10· 3m+
9 · 2m- 4
13.
7
-
12 4x + 1 . 30x + 60
• 5x + 10
2x - 2
36)
(x 2
15.
-
2
1) · 3xx ~ 3
(page 536)
Example 4
(page 536)
Find the reciprocal of each expression.
2
2
16. X +
17· 2d-6d
1
- 5
18. c 2 - 1
19. s + 4
Divide.
21 3t + 12 -'- t + 4
22. ----ro-
23. x - 3
24. x 2 + 6x + 8
7
2
25 • 2n2 - 5n - 3 _,_. 4n + 5
4n - 12n - 7
2n - 7
27. 1~1 ~ ~~1
(k
3 - x
2
7
6
Example 5
y - 4 . 4-y
-;- -5-
. x+3
20" x-1
x+4-;-x+4
26. 3x: 9
7
(x
+ 3)
•
5t
lOt
.
x2 + x - 2
7
..:L±.__±_
2x + 4
+ 11)
(page 536)
0
Apply Your Skills
28. x2 + lOx - 11
2
x + 12x + 11
7
(x _ 1)
Multiply or divide.
29. t 2 + 5t + 6 . t 2 - 2t - 3
t - 3
t 2 + 3t + 2
6t 2 - t - 15
31. ]t 2 - 28t
2t - 5t - 12
49t 3
33 x 2 + x - 6 _,_ x 2 + 5x + 6
• x 2 - x - 6 . x 2 + 4x + 4
2
30 • c2 + 3c + 2
c - 4c + 3
32 •
7
c + 2
c - 3
5x2 + lOx - 15 _,_ 2x 2 + 7 x + 3
2
.
2
5 - 6x + x
4x - 8x - 5
2
2
34 ( x2 - 25)( x2 + x - 20 )
" x - 4x x + lOx + 25
35. Error Analysis In the work
shown at the right, what
error did the student
make in dividing the
rational expressions?
2
36. Critical Thinking For what
values of x is the expression
2x 2 - Sx - 12 . - 3x - 12
-;2
6x
x - 16
undefined?
37. Write two expressions. Find the products.
Lesson 11-2
Multiplying and Dividing Rational Expressions
537
The formula below gives the monthly payment m on a loan when you know the
amount borrowed A, the annual rate of interest r, and the number of months of
the loan n. Use this formula for Exercises 37-39.
A(fz)(1 +
tzr
m = (1+12
r )n
- 1
38. What is the monthly payment on a loan of $1500 at 8% annual interest for
18 months?
39. What is the monthly payment on a loan of $3000 at 6% annual interest for
24 months?
40. Suppose your parents want to buy the house shown at the left. They have
$20,000 for a down payment. Their mortgage will have an annual interest rate
of 6%. The loan is to be repaid over a 30-year period.
a. How much will your parents have to borrow?
b. How many monthly payments will there be?
c. What will the monthly payment be?
d. How much will it cost your parents to repay this mortgage over 30 years?
FOR SALE
$200,000
\.::1+\::.1
Find the volume of each rectangular solid.
41.
42.
x-5
3x+ 2
2m+ 4
m
J-------/
/
/
x-2
x2
m3
+ 2x-
35
m2- m- 6
m2 + m- 2
43.
m2
+ m-
12
44.
2a- 1
2a2+7a-15
r2- 9
r2- 6r + 9
;----
/
/
4a- 6
3a- 1
/
/
/
r+ 2
r2- 1
3a2 +Sa- 3
2a2 +Sa- 3
45. Writing Robin's first step in finding the product ~ · w5 was to rewrite the
expression
a
a
Challenge
Multiply or divide. (Hint: Remember that
46.
Homework Video Tutor
Visit: PHSchool.com
Web Code: bae-11 02
538
5
as~ · ~ • Why do you think Robin did this?
Chapter 11
48
3m3 - 3m • (6m2
4m2 + 4m- 8
+ 12m)
5x2
. .:. . 5xy - 25x
• y 2 - 25 . y 2 - lOy + 25
50.
3m
m- 1
6m 2
m- 2
Rational Expressions and Equations
51.
~ = ~ + ~.)
d
47
t 2 - r2
. =---:t2=-+-'----=3t::..:. _r_+'------=2r-=-2
2
2
• t + tr - 2r
t 2 + 2tr + r 2
2a 2 - ab - 6b 2 . 2a 2 - 7ab + 6b 2
49. 2
2 -;2
2b + 9ab - Sa
a - 4b
3x
x2 - 1
6
x 2 - x- 2
52.
w- 3
w2 - 4
w2 - 9
w-2
I
I
..-:----
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Alg113.0
53. What is the area of a square with a side length of 4x2y 4 units?
®
Alg1 6.0
®
2xy2
CO
8x4y16
®
16x4y8
16x4y16
54. Which graph shows a line with the same y-intercept and half the slope of
y = X - 2?
ty
®
;(
CO
tY
2
X
-2
®
®
X
x;_:~IVIi.xe~t:iiev~if!w
Lesson 10-2
for
Help
Lesson 9-2
,.., _,.
Assume a and b are legs of a right triangle, and c is the hypotenuse. Find the length
of the missing side of each right triangle. If necessary, round to the nearest tenth.
55. a = 2, b = 8
56. a = 3.1, b = 4.3
58. a= VIO,b =VITI
59. a= ~,b = 2
l
57. a= Vl,c =
60. a = 2 ~, b =
V32
6~
Graph each function. Label the axis of symmetry and the vertex.
61. y = x 2
+ lOx - 2
62. y = x 2 - lOx - 2
r 0 -cliecl<point Qiiiz l
63. y = 2x 2
+x +5
[essons 1·1:·1...ttirough- l1: 2
Simplify each expression.
1•
Sb - 25
10
4
10a 4 -
5a 2
'
15a 2
3
2 -36k 4
7m- 14
3• 3m- 6
• 48k
5•
2z2 - llz - 21
2
z - 6z - 7
2
6 • 4c 2 - 36c + 81
4c - 2c - 72
Multiply or divide.
6. c + 4
7• 3cSc
c- 2
2 - 4 . x 2 + 7 x + 12
9 • xx+3
x-2
nline Lesson Quiz Visit: PH School. com, Web Code: baa-11 02
5 . .:. . 3z + 15
8• z +
z .
4z
2
10. 2v +A?v - 3 -:-(12v 2 - 6v)
539
Dividing Polynomials
~ California Content Standards
10.0· Add, subtract, multiply, and divide monomials and polynomials. Solve multi-step problems,
including word problems, by using these techniques. Master
What You'll Learn
for Help
@ Check Skills You'll Need
• To divide polynomials
... And Why
To find the length of a
rectangle, as in Example 3
Lessons 8-1 and 8-3
Write each polynomial in standard form.
1. 9a - 4a 2 + 1
2. 3x 2 - 6 + 5x - x 3
3. -2 + 8t
Find each product.
4. (2x + 4)(x + 3)
6. (3a 2
5. (- 3n - 4)(n - 5)
+ 1)(2a - 7)
Dividing Polynomials
To divide a polynomial by a monomial, divide each term of the polynomial by the
monomial divisor.
Dividing a Polynomial by a Monomial
Divide 8x 3
(8x 3
+ 4x 2
+ 4x 2
-
12x by 2x 2 .
- 12x) -:- 2x 2
= (8x 3 + 4x 2
1 Multiply by the reciprocal of 2x 2.
- 12x)2
2
=8x3
- +4x-2 -12x
2x2
2x
2x 2
=
@CA Standards (heck
•
+
2x 0 -
= 4x + 2-
rf ' Divide.
(
\.!.1
4x 1
a.
3m 3
_
6
m2
+ m) -:- 3m 2
2x
¥
¥
Use the Distributive Property.
Use the division rules for
exponents.
Simplify.
b. (8t 5 + 16t 3 - 4t 2 + 2t) -:- 4t 2
The process of dividing a polynomial by a binomial is similar to long division. For
example, consider dividing 737 by 21.
35
21h37
63
107
105
2
1. Divide: 21 can go into 73 about 3 times.
2. Multiply 3 x 21 and then subtract from 73.
3. Bring down the 7. Divide: 107 + 21 ~ 5.
4. Multiply 5
x
21 and then subtract from 107.
5. The remainder is 2.
737 -:- 21 = 35
il
You can summarize the process for long division as
"Divide, multiply, subtract, bring down, and repeat as necessary."
In the division above, the answer is written as a mixed numbe!: 35 2~ means 35
In dividing polynomials, write the answer as quotient + re~-~i~-~er.
540
Chapter 11
Rational Expressions and Equations
+
:£1 .
I
When the divisor and dividend are in standard form, divide the first term of the
dividend by the first term of the divisor to find the first term of the quotient.
_
Dividing a Polynomial by a Binomial
Divide 2y 2 + 3y - 40 by y + 5.
Step 1 Begin the long division process.
Align terms by their degrees.
So put 2y above 3y of the dividend.
~
2y
y
+
5J2y 2
2y 2
+
+
lOy
Divide: Think 2y 2 + y = 2y.
Multiply: 2y(y + 5) = 2y 2 + 1Oy. Then subtract.
-7y- 40
Bring down -40.
3y - 40
Step 2 Repeat the process: divide, multiply, subtract, and bring down.
2y - 7
y
+
5J2y 2
2y 2
+ 3y - 40
+ lOy
-7y- 40
-7y- 35
-5
Divide: -7y + y = -7.
Multiply: -7(y + 5)
The answer is 2y - 7 + Y --_; 5 , or 2y - 7 - Y ~
@CA Standards Check
(2) Divide.
a. (2b 2
-
b -
3)
(b
7
=
-7y - 35. Then subtract.
The remainder is -5.
+ 1)
5.
b. (6m 2 - 5m - 7)
7
(2m
+ 1)
When the dividend is in standard form and a power is missing, add a term
of that power with 0 as its coefficient. For example, rewrite 4b 3 + 5b - 3
as 4b 3 + Ob 2 + 5b - 3.
Dividing Polynomials With a Zero Coefficient
The width and area of a rectangle are shown in the
figure. What is an expression for the length?
w
= (2b- 1) in.
Divide the area by the width to find the length.
2b - lJ4b 3
+
2b 2
Ob 2
+
+
b + 3
5b - 3
Rewrite the dividend with Ob 2.
lA = (4lJ3 + Sb- 3) in. 2
4b 3 - 2b 2
2b 2 + 5b
2b 2 - b
6b - 3
6b - 3
0
An expression for the length of the rectangle is (2b 2 + b + 3) in.
@CA Standards Check
J) Divide.
a. ( t 4 + t 2 + t -
3)
7
(t - 1)
b. (c 3 - 4c + 12)
Lesson 11-3
7
(c
+ 3)
Dividing Polynomials
541
To use the process for long division, write any divisor or dividend in standard form
before you begin to divide.
To review standard form,
see Lesson 8-1, Example 2.
Reordering Terms and Dividing Polynomials
Divide -3x + 4 + 9x 2 by 1 + 3x.
Rewrite -3x + 4 + 9x 2 as 9x 2 - 3x + 4 and 1 + 3x as 3x + 1. Then divide.
3x - 2
3x + lJ9x2 - 3x + 4
9x 2 + 3x
-6x + 4
-6x- 2
6
The answer is 3x - 2 + 3x 6+ 1·
ciCAStandardsCheck
~
4) Div(idoe.- 1 + 8x2)-:- (1 + 2x)
a. 1 x
Summary
b. (9 - 6a 2 - lla) -:- (3a - 2)
Dividing a Polynomial by a Polynomial
Step 1
Arrange the terms of the dividend and divisor in standard form.
Step 2
Divide the first term of the dividend by the first term of the divisor.
This is the first term of the quotient.
Step 3
Multiply the first term of the quotient by the divisor and place the
product under the dividend.
Step 4
Subtract this product from the dividend.
Step 5 Bring down the next term.
Repeat Steps 2-5 as necessary until the degree of the remainder is less than the
degree of the divisor.
For more exercises, see Extra Skills and Word Problem Practice.
:' Standards Practice
Practice by Example
Example 1
(page 540)
for
Help
Example 2
(page 541)
Alg1 10.0
Divide.
1. (x 6 - x 5 + x 4 ) -:- x 2
2. ( l2x 8 - 8x 3 ) -:- 4x 4
3. (9c 4 + 6c 3
4. (n 5 - l8n 4 + 3n 3 ) -:- n 3
-
c2 )
-:-
3c 2
5. (8q 2 - 32q) -:- 2q 2
6. ( -7t 5 + l4t 4 - 28t 3 + 35t 2) -:- 7t2
7. (x 2 - 5x
8. (2t 2 + 3t - 11) -:- (t - 3)
+ 6) -:- (x - 2)
9. (n 2 - 5n + 4) -:- (n - 4)
11. (3x 2
-
1Ox + 3) -:- (x - 3)
13. (4a 2 + 6a + 12) -:- (a+ 2)
542
Chapter 11
Rational Expressions and Equations
10. (y 2 - y
+ 2) -:- (y + 2)
12. (- 4q 2 - 22q
14. (2m 2
+ 12) -:- (2q + 1)
+ 13m + 15) -:- (m + 4)
Example 3
{page 541)
Divide.
(st 2
-
17. (3b 3
-
15.
19.
soo)
(t 3 -
(t
7
10b 2 + 4)
6t -
4)
16. (2w 3 + 3w - 15)
+ 1o)
(3b - 1)
7
(t + 2)
7
18. (c 3 - c 2 - 1)
20.
(n 3 -
7
7
(
25n - 50)
(w - 1)
c - 1)
(n + 2)
7
21. The width of a rectangle is (r - 5) em and the area is (r 3 - 24r - 5) cm2 .
What is the length?
22. The base of a triangle is (c + 2) ft and the area is (2c 3 + 16) ft 2 .
What is the height? (Hint: The formula for the area of a triangle is A =
Example 4
{page 542)
Divide.
23. ( 49
+ 16b + b 2)
+ 2x 3
27. ( -13x
33. (56a 2
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Web Code: bae-11 03
-35. (3k 3
0.9k2
37. ( -2z 3 - z
39.
7
7
7
1.2k)
7
+ z2 + 1)
7
-
(x - 3)
28. (6 - q
+ 3t - 1)
7
9)
-
+ 3q 3
(4 + a)
7
(4 + t)
7
- 4q 2 ) 7
30. (c 3 + llc 2 - 15c +
8)
(q - 2)
c
7
3Z. (4y + y 3 -
7)
+ 1)
34. (5t 4 - 10t 2
+ 6)
7
(t + 5)
3k
36. (-7s + 6s 2 + 5)
7
(2s
(z + 1)
38. (6m 3 + 3m + 70)
(64c 3 - 125) 7 (5 - 4c)
41. (2t 4 - 2t 3
6 + 3a)
26. (4t + t 2
2x 3
(2a
-
+ w)
(b - 1)
7
+ 4a - 12)
-
(72
7
6 - x 2)
29. (6x 4 + 4x 3 - x 2 )
31. (8b + 2b 3 )
Homework Video Tutor
-
24. (a 2
+ 4)
(b
7
+ 14 + 10w 2 )
25. (39w
Apply Your Skills
1bh.)
40.
(2t 3 + 1)
(y - 5)
7
(21 - 5r 4 - 10r 2
42. (z 4 + z 2
-
2)
7
(m
+ 2r6)
(z
7
+ 3)
+ 4)
7
(r 2 - 3)
+ 3)
43. a. Write a binomial and a trinomial using the same variable.
b. Divide the trinomial by the binomial.
44. The volume of a rectangular prism is 2x 3
+ 5x2 + x - 2. The height of the
prism is 2x - 1, and the length of the prism is x + 2. Find the width of the prism.
45. Writing Suppose you divide a polynomial by a binomial. Explain how you
know if the binomial is a factor of the polynomial.
46. The volume of the rectangular prism shown at the
right is m 3 + 8m 2 + 19m + 12. Find the area of
the base of the prism.
47. a. Find (d 2 - d
b. Find
(d 3 -
c. Find (d 4 -
+ 1) 7 (d + 1).
+ d - 1) 7 (d + 1).
3
d + d 2 - d + 1) 7 (d + 1).
m+3
d2
I
I
d. Predict the result of dividing
d 5 - d 4 + d 3 - d 2 + d - 1 by d + 1.
e. Verify your prediction by dividing the polynomials.
48. Critical Thinking Find the value of kif x + 3 is a factor of x 2
-
x - k.
49. a. Solve d = rt for t.
b. Use your answer from part (a) to find an expression for the time it takes to
travel a distance of t 3 - 6t 2 + 5t + 12 miles at a rate oft + 1 miles per hour.
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543
G
Challenge
Divide.
50. (4a 3b 4 - 6a 2b 5 + 10a 2b 4 )
+ 7xy - 2y 2 )
51. (15x 2
7
7
2ab 2
(5x - y)
52. (90r 6 + 28r 5 + 45r 3 + 2r 4 + 5r 2 )
53. ( 2b 6
+
2b 5 - 4b 4
+
b3
+
(9r + 1)
7
8b 2 - 3)
(b
7
3
+ 2b 2
- 1)
For California Standards Tutorials, visit PHSchool.com. Web Code: baq-9045
Alg110.0
54. Which of the following expressions equals (3x 3 - 4x - 1)
3x2
®
3x2 -3x-1
~
Alg110.0
63x - 7 + -x +1
r7\\
-
©
3x2
®
3x2 - 7x + 8
3x - 7 -
-
55. Which of the following must be true for (x 2 + 2x + 1)
7
(x + 1)?
7
x
8
+
1
(x + 3)?
I. The remainder is negative.
II. The dividend is in standard form.
III. The quotient is larger than the divisor for the positive values.
® Ionly
© Iandii
® II only
® ~I and III
Alg1 6.0
56. Which equation best represents the function shown on
the graph at the right?
= 3x- 1
©
y
ix - 1
®
y =
®
y
®
y =
Lesson 11-2
-ix + 1
n2
+ 7n
n -
-
1
8 .
n
59. 3c2 - 4c - 32
2c 2 + 17c + 35
Lesson 9-3
544
Chapter 11
s)
;,,;})·;y · J.;;<.·
Multiply or divide.
57.
Lesson 9-8
2
= -3x + 1
'EiJtrn11RA!ft a',''·i' ;;:¥t)'":p;~M~;t.?f"Mi9¥5 ,.:,;.;fttrf ;~·, . !.'·;,:;.~i:.
for
Help
y
2
n2 - 4
+ 6n -
7
c - 4
c + 5
58.
16
2 6t
2
6t 2
- 30t
+ 35t + 11
18t 2
2t - 53t - 55
60 x 2 + 9x + 20 _,_ x 2 + 15x + 56
• x 2 + 5x - 24 ·
x 2 + x - 12
Determine whether the graph of each quadratic function intersects the x-axis in
zero, one, or two points.
61. y = x 2 + x + 1
62. y = x 2 + 2x + 1
63. y = x 2 - 8x - 7
64. y = - 3x2 + 4x + 5
65. y = 2x 2 + 5
66. y = 9x 2 - 7x + 144
Find the value of each expression. If the value is irrational, round to the
nearest hundredth.
67.
\128.9
68.
70.
V4000
71.
Rational Expressions and Equations
v'289
v'40
69. -\1161.29
72.
vT69