Download Ch 5

Exponents
and Polynomials
M
Ost people don't have thousands of dollars to
buy expensive things like houses or cars. To buy a house, you go
to a bank and apply for a loan called a mortgage. You need to
borrow enough money to pay for the cost of the house. That is
called the "principal." Along with the principal, you pay the bank
extra money, or interest. Bankers use an algebraic formula to
divide the total mortgage into monthly payments. Each month,
the interest you owe decreases and the principal you're paying off
increases. This kind of calculation is a polynomial. Polynomials
involve doing several arithmetic functions using exponents.
In Chapter 5, you will examine exponents and polynomials.
Goals for Learning
 To recognize and use exponents in computations
 To identify the benefit of using scientific notation in
some calculations
 To define, name, and solve polynomials
12
3
In Chapter 1, you learned how to multiply (.0)
X4) = (X' x • X) (X' x • x • X) = xl.
In other words, (x3) (X4) = X3 + 4 = X7.
(X4): ( X
3
)(
Rule The general rule is (xn)(xm) = xn + '". To multiply terms with
exponents, add the exponents.
You can use what you know about exponents to find
the value of (X2)4:
4 times
2
2
2
,,-...A-..
(X )4 can be written as (X )(X )(X2)(X2) = x8. In other
words, (X2)4 = x2• 4 = X8.
Rule The general rule is (xn)m = xn • m.
To raise a power to a power, multiply the exponents.
Suppose you need to simplify X7 -;- .0.
You can rewrite the problem as
xl _'x·/-/-x·x·x·x _ x·x·x·x=
:(3
,;t.,X.,x
0.
1
x7
In other words, :(3=X7 - 3 = 0.
Rule The general rule is xn -i- xm = xn - "',
To divide terms with exponents, subtract the exponents.
Important note This is only true when x
124
Chapter 5 Exponents and Polynomials
*- O. % is undefined.
, ,
............
I
Raising a number to the zero power is a special case.
Writing About
Mathematics
B_ = ~ =5 25 - 5 = 20 = 1
32
Any number (except
zero) raised to the
power of zero is 1.
For example,
3° = 1. Tell why.
2
or ~ = 2· 2 • 2 • 2 • 2 = 1
2•2•2•2•2
25
What is the value of xo, when x '" O?
You already know that 1 = X2
x2
and that 2'
x2
= x2 - 2 = xo. x
You can put these two statements together:
x2
1 = -2X = x2-2 = Xo so 1 ' = xo
Rule
•
Xo = 1 when x'" O.
You can use what you know about multiplying and
dividing exponents to find solutions to problems. (22)2 =
22• 2 = 24 = 2 • 2 • 2 • 2 = 16
Exercise A Show why these statements are true.
1. (32)3 = 729 2~
(22)5 = 1,024
3.
(X3)4 = X12
4.
(y5)3 = y15
5.
(m4)4 = m16
6.
[(x + y)2]3 = (x + y)6
7.
[(2x + 3y)4]2 = (2x + 3y)8
Exercise B Simplify each expression.
8. (35) -7- (33)
44
9.
43
(x + y)5 -7- (x + y)2, (x + y)
13.
XS
10·4,x*0
x
3q)6
(2p + 3q)2'
14.
(2p + 3q)
*- 0 (2p +
*- 0
11. y5 -7- y3, y*-O
Exercise C Show two ways to simplify each expression.
(4x + 2y)5 -7- (4x + 2y)\ (4x + 2y)
18.
q)3
19.(
p+q
p4
20.
4'P
p
)3'(p
*- 0 (p +
+ q) *- 0
*- 0
Exponents and Polynomials Chapter 5 125
Negative
exponent
For any nonzero
integers a and n,
a-n = _!_
The number patterns below show how to use negative
exponents to show numbers less than 1.
103
= 1,000
)-;.- 10 gives
23
=8
) -;.- 2 gives
102
= 100
)-;.- 10 gives
22 =4
) -;.- 2 gives
101
10
)-;.- 10 gives
21 =2
)-;.- 10 gives
2° = 1
an
10°
1 1)
2-1
=1O=1QT
-i-
) -;.- 2 gives
-"2-2"1
10 gives
1 1)
2-2
= 1 00 = -,--oz
-i-
t
= 4 Rewrite
= ~3 )
Rewrite 1~4 with a negative exponent: 1~4 = 10-
1
= 1,000 = 103
y)5with
a negative
and so
on
1
(x + y)5 - X
_(
+Y
exponent, (x + y) =I=- 0:and so on
)-5
You can also use the rule for exponents in division to find
negative exponents. Remember 3° = 1.
l_=~=30-2=3-2
32
1 XO
32
•
-x3= - =x3XO - 3 = x-3
126
Chapter 5 Exponents and Polynomials
2 gives
-i-
2 gives
-"4-2"2
1 10
2-3
-i-
_1_1)
10 gives
10-3
(x
2 gives
_1_1)
10-1
10-2
) -i-
1 j J! j ! ! ! ! j 1 !
Writing About
Mathematics
= x5 -7 = x-2
or
Explain why
~2can be rewritten
as 3-2•
(2x + 3y)8 = (2x
(2x+3y)11
+ 3y)8-11 = (2x + 3y)-3
Write (2x + 3y)-3 with a positive exponent:
(2x + 3y)-3 = (2x + 3y)-3 • 1 = (2x + 3y)-3 • (2x + 3y)3 (2x + 3y)3
_ (2x + 3y)-3 • (2x + 3y)3 _ (2x + 3y)-3 + 3
(2x + 3y)3
(2x + 3y)3
_ (2x+3y)° _
1
- (2x + 3y)3 - (2x + 3y)3
Exercise A Rewrite using a negative exponent.
45
1.
6.
49
1
2.
1023
1
7.
53
152
1
3.
8. 143 -i- 145
105
2033
4.
25
--:-
210
9.
2037
s.
lOlO --:- 1020
10. 5,2802 --:- 5,2803
Exponents and Polynomials Chapter 5 127
Exercise B Rewrite using a negative exponent.
x"
11. 7,x-=l- 0
x1
12. -3' m -=I- 0
16.
m
13. (
14.
15.(
1
x+ 2y
)4,(x+2y)-=l-O
(x + 3y)3 7 (x + 3y)5,
(x + 3y) -=I- 0
(x + y)3 (
x+y
)6' X +
(2x + 3y)2 7 (2x + 3y)8,
(2x + 3y) -=I- 0
16. (-3m -9)3 7 (-3m - 9)6,
(-3m -9) -=I- 0
x3
18·7,x-=l-O
x
)
Y -=I- 0
Exercise C Rewrite using a positive exponent.
19. 3-2
23. 2y-6
20. 10-2
21. 1022. x-4
23
24. (4p+3q)-3
25. 4-2
On Opposite Ends of the Ruler
Particle physicists are scientists who study
things smaller than atoms-particles
measuring as small as 10-16 meters! They have found
quarks and leptons, and they suspect even smaller particles exist.
Astrophysicists study things in the largest scale you can imaginethe
universe. They have found that less than 10 percent of the mass of the
universe consists of the kind of matter we can see. The last century
revealed much about our universe. Just imagine what this one will bring!
128
Chapter 5 Exponents and Polynomials
Lesson 3
i;J XE)(ponents and Scientific Notation
Scientific
notation
A number written as the
product of a number
between 1 and 10 and a
power of ten.
Any number in
scientific notation
= (1 :S X < 10) (lO")
Scientists and researchers often need to record and work with very
large or very small numbers. To make writing these numbers
easier, they have developed a way of writing these numbers called
scientific notation.
Write 394.74 in scientific notation. 394.74
=
3.9474(102)
Step 1 Move the decimal point so the number is between 1 and
10.
394.74 U
Step 2 Write the new number and then count the number of
places the decimal point moved. In this example, the
decimal point moved 2 places to the left.
3.9474
Step 3 Multiply the number from Step 2 times 10 raised to the
number of places the decimal point moved.
3.9474(102)
,
.
~
Use a positive
exponent if the
decimal point
moved to the left.
Use a negative
exponent if the
decimal point
moved to the right.
Write 0.0003947 in scientific notation.
Step 1 Move the decimal point so the number is between 1 and
10.
0.0003947
~
Step 2 Write the new number, and then count the number of
places the decimal point moved. In this example, the
decimal point moved 4 places to the right.
3.947
Step 3 . Multiply the number from Step 2 times 10 raised to the
number of places the decimal point moved. 3.947.10-4
Exponents and Polynomials Chapter 5 129
; J J1 ! J! i !
Writing About
Mathematics
; J!
Find a use of
scientific notation in
a science book or an
encyclopedia. Write
a sentence about
what you found and
share your findings
with other students.
Exercise A Write each number in scientific notation.
1. 186,000
6. 0.002010
2. 0.00563
7. 0.0000000001
3. 276,000,000,000
8. 1,000,000,000
4. 0.0156
9. 935,420,000
5. 1,342.54
10. 0.00000305
Exercise B Rewrite these numbers so they are no longer in scientific
notation.
11. 3.28· 10-3
16. 5.42· 10-6
12. 5.42· 106
17. l.6· 10-27
13. l.86(102)
18. l.1122(10-4)
14. 2.71 (10-8)
19. l.1122(104)
15. 5.280(103)
20. 3.1· 10-4
You can use a calculator to translate numbers into scientific
notation. Some calculators have a function or mode that will
translate the number immediately.
Translate 250 into
scientific notation.
Select scientific mode
Enter number 250
Press ~ Enter I
Some calculators translate
numbers using these steps:
Enter 250
Press ~ I
[ill El
Read 2.500002
Press El
Read 2.500002
Read the display as 2.5(102)
Exercise C Use a calculator to check your answers in Exercise A.
130
Chapter 5 Exponents and Polynomials
Scientific notation is useful for multiplying or dividing very
large or very small numbers.
.'
Find (13,000,000)2.
Step 1 Write the number in scientific notation.
13,000,000 = 1.3 • 107
(13,OOQ,OOO)2 = (1.3 • 107)2
Step 2 Change the order of the factors to make the
multiplication easier.
(1 .3 • 107)2 = (1.3 • 107)(1.3 • 1 07) =
1.3 • 1 .3 • 1 07 • 1 07
=
(1.3)2(107)2
Step 3 Complete the multiplication, using the rule for
raising exponents to a power.
(1.69)(107•2) = (1.69)(1014)
Step 4 Check that the product is written in
scientific notation.
The answer is 1.69(1014).
Find the product of 0.0000006 •
32,000,000 • 0.0043.
Step 1 Write each number in scientific notation. (6.0.
10-7)(3.2 • 107)(4.3 • 10-3)
Step 2 Use the commutative property to change the order of the
factors.
(6.0 • 3.2 • 4.3)(10-7 • 107 • 10-3)
Step 3 Complete the multiplication, using the rule for
multiplying with exponents.
(82.56)(10-7+7-3) = (82.56)(10-3)
Step 4 Write the product in scientific notation.
(8.256)(10)(10-3) = 8.256(10-2)
Exponents and Polynomials Chapter 5 131
Find the quotient of 9,250,000 -;- 25,000.
Step 1 Write each number in scientific notation. 9.25.106
-i-
2.5 • 104
Step 2 Rewrite the division as a fraction.
9.25 • 106= 9.25 • ~ = (9
2.5.104
2.5
104
25 -i- 2 5)(106 -;- 104)
•
•
Step 3 Complete the division, using the rule for dividing with
exponents.
(3.7)(106 - 4) = (3.7)(102)
Step 4 Check that the product is written in scientific notation.
The answer is 3.7(102).
Exercise A Find the products. Write your answer in
scientific notation.
1. 1.4(103).6.3(10-4)
2. 8.1(1014).9.0(10-6)
3. (4.01. 102)3
4. (3.4. 102)(1.3. 105)(2.54. 10-6)
5. 52,000,000· 706,000
6. (11,000,000)2
7. (0.00008). (640,000,000)
8. (350,000). (1,200) • (16,000,000)
9. (0.00645). (0.00004302) • (0.000000035)
10. [(2000)3. (50,000)2]2
132
Chapter 5 Exponents and Polynomials
Exercise B Find each quotient. Write your answer in
scientific notation.
16. 350,000 -i- 1,400,000
12. (7.62. 10-2) -i- (2.54. 106) 17. 0.00008 640,000,000
1,200
18. ----'----
240,000,000
19.
15. 545,000,000 --:- 100,000
20.
0.000645
0.005
50,0002
0.00001
Exercise C Use a calculator with a scientific mode to check
your answers to Exercises A and B.
ifechno!!?9Y Connection
Smashing Atoms
Scientists use accelerators-atom smashers-to release the
parts of atoms called quarks and leptons. One accelerator in the United
States uses 1,000 superconducting magnets. Each magnet weighs about
20 tons. The magnets steer bunches of protons-parts of atoms-around
the accelerator, or ring, as the scientists call it. Each bunch contains
more than one trillion protons. No wonder scientific notation was
invented! How else could scientists work with such huge numbers?
Exponents and Polynomials Chapter 5
133
Polynomial
An algebraic expression
made up of one term or
the sum or difference of
two or more terms in the
same variable
You have already worked with algebraic expressions such as
3a - b + 5 - 2a. In this lesson, you will be introduced to
algebraic expressions known as polynomials.
Monomial
A term that is a
number, a variable, or the
product of a number and
one or more variables
Degreeofa
polynomial
Greatest power of the
variable
-one term
monomial
3x2
two terms
binomial
3x2
three terms
trinomial
many terms
polynomial
+X
+ X+ 1
3
x + 2X2 - X + 5
Some algebraic expressions are not polynomials.
x-4 + x2 + 3
1
has a negative exponent
is not a sum or difference
-2
x +3
5
is not a sum, has a negative exponent
-3
x-
2y3 + x2 +
X
+1
3
has more than one variable
has a variable in the denominator
x2
Polynomials can be named by their terms.
trinomial (or polynomial) in y
x2 + 1
x2 + X +
1 y2+y+l
polynomial in z
2z3 +
binomial (or polynomial) in x
trinomial (or polynomial) in x
Z2
+ z+ 1
The greatest power of the variable is called the degree of a
polynomial.
6
degree 0 (The degree of a constant is 0.)
x-3
degree 1
2
x + X+ 1
degree 2
3
2
x + 4x + 1 degree 3
134
Chapter 5 Exponents and Polynomials
The terms in a polynomial can be arranged in any order.
However, in standard form, they are arranged left to right, from
greatest to least degree of power. Always place the terms in
standard form before using a polynomial in a computation.
Standard form
_-JTangement of
ariables from left to 10ht, from greatest to
.east degree of power
erclse A Name each expression by the number of its terms.
Lse monomial, binomial, trinomial, and polynomial.
1. y2 + 1
6. 2x:5 + 0 + x + 23
2. n2 + n + 1
7. 7y4 + 4y + 8
3. 5x2 + 2x - 4
8. 7a3 + a2
4. 5z4 +
9. b7 + b6 +
5.
+ Z+ 1
2p3 + p2 + P _ l._
Z3
10.
lJ5
+ b4 + b 3 + b 2 + b
blO
3
Exercise B Give the degree of each polynomial.
11. 3n2 + 2n + 1
12. 3w
2
16. 2x:5 + X4 + 0 + x + 23
+ 2w + w - 31
17.
z7
+ 5z4 + Z3 + z + 432
13. 5x4 + 20 + 54
18. 7b4 + 7b3 + 4b2 + 8
14. 5a5 + a3 + a + 1
19. a3 + a2 + a
1
2s3
+ S2 + s --
20. F +2b6 +3b5 +4b4 +5b3 +6b2 +7b
15.
2
Exercise C Write one polynomial for each description.
21. a trinomial in x, degree 3
22.
a polynomial in y, degree 4
24. a polynomial in z, degree 3
25. a monomial in r, degree n
23. a binomial in b, degree 5
Exercise D Tell why these expressions are not polynomials.
26.
27.
28.
3n-2 + 2n + 23
2
3w -;- 2w
29 2
.X
'
30. x3 + 5y2 + X - 5
5x
4
+
_1
_
3x'l
Exponents and Polynomials Chapter 5 l35
You can find the sum of two or more polynomials by adding like
terms.
Add (5X4 +
x3 _2X2
+ 7x - 5) and (-2x3 + x2 - 5x + 3).
Rewrite the expression and line up like terms.
Step 1
5x4
+
+
x3
2x3,
+
2X2 +
x2
7x
5x
2X2 +
x2
2
x
+
7x
5x
2x
+
5
3
+
5
3
Add like terms.
Step 2
5x4
+
5x4
-
+
x3
2x3
x3
+
2
You can find the difference of two or more polynomials by
subtracting like terms.
Subtract (- 2x3 + x2 - 5x + 3) from
(5X4+ x3 - 2X2 + 7x - 5).
Step 1 Remember that to subtract, you add the opposite. -(-2x3
+ x2 - 5x + 3) is equal to
(-1)( - 2x3 + x2 - 5x + 3)
=
2x3 - x2 + 5x - 3
Step 2 Rewrite the expression and line up like terms.
5x4 +
x3
+
2X2 + 7 x
2x
5
x + 5x
3
2
3
Step 3 Add like terms.
2X2 +
+
x2
+
3x2 + 12x
136
Chapter 5 Exponents and Polynomials
7x
5
5x
8
3
rclse A Find each sum.
1. (2y5 + y3 + 7y + 33) and (4y6 + y3 - 4y2 - 7y - 5)
2. (2X4 - 4X3 - 15x2 + 21x + 4) and (4x4 + 2x3 + 17)
3. (b4 + b3 -2b2 + 7b - 5) and (-3b4 - b3 -2b2 -2b)
4. (m4 + m2 - 5) and (m3 + m + 5)
5. (4x4 + 7x3 + 15x2 + 4) and (4x3 + 2x2 + 17x)
6. (2b5 + 3b4 - 4b3 + 7b2) and (-2b - 12)
7. (-4x7 - 6x5 - 7x3 - 9x - 2) + (-xl + 6x6 + 2x2 + 8)
,
8. (5m4 + 2m2 - 5m) and (5m3 + 2m + 10)
9. (x7+x5-3x3) + (x7-6x6+8x - 2) + (-2x2+8x - 14)
10. (7y5 + 8y2 + 3) + (7y5 + y3) + (-4y2 - Y - 3)
ercise B Find each difference. Remember to add the opposite.
11. (2 y5 + y3 + 7 Y + 33) - (4 y6 + y3 - 4 y2 - 7 Y - 5)
12. (2X4 - 4x3 -15x2 + 2lx + 4) - (4x4 + 2x3 + 17)
13. (b4 + b3 - 2b2 + 7b - 5)- (-3b4 - b3 - 2b2 - 2b)
14. (m4 + m2 - 5) - (m3 + m - 5)
15. (x7 + x5 - 3x3 + 8x - 2) - (x7 - 6x6 -3x3 + 8x - 14)
16. Subtract (7y5 + y3 - 4y2 - y -3) from (7y5 + 8y2 + 3)
17. Subtract (4x3 + 2X2 + 17x) from (4x4 + 7x3 + 15x2 + 4)
18. Subtract (-2b - 12) from (2b5 + 3b4 - 4b3 + 7b2)
Exercise C Follow the directions.
19. Franco is remodeling his kitchen.
He is going to put baseboards around
the perimeter of the room. Write an
expression that shows the perimeter of
Franco's kitchen.
~
L_jX+5
20. For an art project, Clarissa is decorating
the lid of a box shaped like a triangle.
She is gluing lace to the perimeter of the
triangle-shaped lid. She wants to know
what the
perimeter of the lid is so that she can cut
the right amount of lace. Write an
expression to help her find the perimeter
of the lid.
1/,<-3
X-~
X2
x+ 1
Exponents and Polynomials Chapter 5
137
You can use the distributive property to multiply monomials and
polynomials .
..
x2(4x5 + X3 - 2X2 + X - 5) =
(X2 • 4x5) + (X2 • X3) - (X2 • 2X2) + (X2 • x) - 5x2 = 4X7 + X5 - 2.0
+ X3 - 5x2
Suppose you must simplify the following expression:
3(a + b) + 2(a + b). You can use the distributive property and
simplify this sum in two ways. Hint: Think of (a + b) as a
single variable as in 3x + 2x = Sx.
3(a
+ b)
r
2(a
+ b)
(3 + 2~(a + b)
3a + 3b + 2a + 2b
S(a + b)
t
t
Sa + Sb
Sa + Sb
You can use the distributive property to multiply
two binomials.
Find the product of (x + 1 )(x - 4).
I
1ft
(x + l)(x - 4)
x(x - 4) + (l)(x - 4)
(x2 - 4x) + (x - 4)
x2 - 3x - 4
Find the product of (x - 1 )(x - 4).
I
I
(x - 1 )(x - 4)
I
•
t
x(x - 4) - (1 )(x - 4)
•
(x2 - 4x) - (x - 4) x2 - 5x
+4
Chapter 5 Exponents and Polynomials
You can use the distributive property to find the product of a
binomial and a trinomial.
Explain in words or
draw a diagram
showing how you
would multiply
Find the product of (x + 4)(X2 + 3x + 1).
(x + 4)(x2 + 3x +1) = x(x2 + 3x +1) + (4)(x2 + 3x + 1) = (x3 + 3x2
+ x) + (4x2 + 12x + 4) = x3 + 7 x2 + 1 3x
(a+b+c)(x+y+z).
+4
Exercise A Find each product.
1. (x + 2)(8x - 2)
6. (2y + 3)(4y3 - 7y - 2)
2. (-8y2 + 3)( -4y - 3)
7. (2X4 - x)(x2 + 2x + 1)
3. (b4 + b3)2
8. (-9x - 2)( -x6 + 2X2 + 8)
4. (m4 + 5)(m3 + m2 + 1)
9. (5m4 + 2m2 - 5m)(2m + l O)
5. (l5x2 + 4)( 4x3 - 17x)
10. (-2b -12) (2b5+3b4-4b3+7b2)
Exercise B Write a polynomial for each problem.
11. Ellie and Terrell are carpeting their
square living room. They need to know
the area of the room to buy the carpeting.
What is the area of the room with each
side measuring
a+2b?
D
13. Ellie and Terrell are also putting new
carpeting in their family room, which is a
rectangle having length = x + 3 and
width = x + 1. What is the area of the
family room?
x+3
a + 2b
12. Before the carpeting is put down in the
living room, Ellie is going to paint the
baseboards. She needs to know the
perimeter of the room to help her
estimate how much paint she will need.
What is the perimeter of the room?
-L___I ___JI x +
14. If their family room had a length twice
that of its actual length, what would the
area of the room be?
14. The kitchen has a width that is three
times the width of the family room but
the same length. What is the area of the
kitchen?
Exponents and Polynomials Chapter 5
13
9
1
The distributive property can be used to divide a polynomial by a
monomiaL
Find the quotient of (12x4 - 8x3 + 4X2) -;- 4.
Step 1 Rewrite the problem.
12x4 - 8x3 + 4x2 4
(dividend)
(divisor)
Step 2 Divide each term of the numerator by the term in the
denominator.
12x4 _ 8x3+ 4x2= __:I2x4 _ ~x3 + '±x2 = 3x4 _ 2x3 + x2
4
4
4
4
4
4
Step 3 Check the answer. In this example, because there is no
remainder (or a remainder of 0), you can check the
answer by multiplying:
• (divisor)
(quotient)
(3X4 - 2x
3
+ x ).
2
(4)
(dividend)
12x4 - 8x3 + 4x2
Find the quotient of (3x3 - 5x2 + 4x) -;- x.
Step 1 Rewrite the problem.
3x3 - 5x2 + 4x x
(dividend)
(divisor)
Step 2 Divide each term of the numerator by the term in the
denominator.
l_~ _ 5x2 + 4x= 3x3 -, _ 5x2 -, + 4x' - ,
x
x
=
3x2 -'5x + 4
x
Step 3 Check the answer by multiplying.
(quotient)
• (divisor)
(3X2 - 5x + 4) • (x)
142
Chapter 5 Exponents and Polynomials
(dividend)
3x3 - 5x2 + 4x
Be sure to watch for + and - signs when dividing polynomials.
-3y4=
_y2 ,
Remember
(-) -0- (-) =
( -) -0-
(+)
+,
(+) =
(-),
-0- (-) = (-).
and
3y2.
check:
-3y4= -3y2.
y2 ,
while
3y4=
_y2 ,
and
(3y2)(_y2)
check: (- 3 y2)(y2)
=
-3y4
= -
3 y4
-3y2.
check: (- 3 y2)( - y2) = 3 y4
Exercise A Find each quotient. Check your work
using multiplication.
(16x2 + 4)
1. 4
(9y3 + 6y - 3)
2.
3
(-32x3 + 24x2 - 16x + 8)
3.
8
3
2
(-32a + 24a - 16a + 8)
4.
-4
5. (7m2 - 7m + 7) -7- 7
6. (-18p3+36p2+9p)-7-9
Exercise B Find each quotient. Check your work
using multiplication.
7.
-'..:( 2:_::6x:_:_3_:+_4::_:x_.L.2)
x
(-21/- 6y4 - 21y)
8.
y
10. (-5m7 -7m2 + 7m) -7- m
11.
12.
(- 25x6 + 14x2 + 9x) -7- - X
(-32a3 + 24a2 - 16a)
.
-4a
Exercise C Find each quotient. Check your work
using multiplication.
13.
(15y5 + y4+
5/ - 17y2 + y)
y
Exponents and Polynomials Chapter 5 143
,;!.,,_;;~ ,sM-" ., ~f>(_U:'
~
Lesson ,,10' f)jDi,vidip,g~,a Polxnomial by a Binomial
-----
Dividing a polynomial by a binomial is similar to the long
division you learned in arithmetic. As in arithmetic, division can
result in a quotient and no remainder or a quotient and a
remainder that is not zero.
Case I
Remainder = 0
Find the quotient of 345 + 15.
Step 1
Divide 15 into 34.
2
2
15) 345
15) 34
Step 2
2
15)345
-_]_Q
Multiply and subtract product.
(2)(15) = 30
23
15) 345
Step 5
Step 2 Multiply and subtract product.
Step 3 Bring down - 6 and divide x into -
3
15)45
Multiply and subtract product.
(3)(15) = 45
45 - 45 = 0
3x.
x-3
x + 2) x2 - x - 6 - (X2
+ 2x)
- 3x - 6
-3
x) - 3x
Step 4 Multiply and subtract product.
~_____:x_:_:----=-3 (- 3)(x+ 2) = - 3x-6 x +
2) x2 - x - 6
- (x2 + 2x)
- 3x- 6
- (- 3x - 6)
Check by multiplication.
(23)(15) = 345
144
x )X2
into 45.
45
23
15) 345
- 30
45
- 45
0
X
x
x + 2) x2 - x - 6
Bring down 5 and divide 15
-_]_Q
Step 4
Step 1 Divide x into x2•
~~_:_:x__ (x)(x + 2) = x2 + 2x
x + 2) x2 - x - 6
x2- X
- (X2 + 2x)
- xL 2x
- 3x
0 - 3x
34 - 30 = 4
4
Step 3
Find the quotient of (x2
- x - 6) -i- (x + 2).
Chapter 5 Exponents and Polynomials
o
Step 5 Check by multiplication.
(x - 3)(x + 2) = x2 - X - 6
Case II
Remainder
*- 0
Find the quotient of (x2
- x - 7) +- (x + 2).
Find the quotient of 346 +- 15.
Step 1
2 15)
346
Step 2
2 15)
346-
_lQ
Divide 15 into 34.
2
15) 34
Multiply and subtract product.
(2)(15) = 30
34 -30 = 4
4
Step 3
23
15)346
-_lQ
46
Bring down 6 and divide 15
into 46.
3
15)
46
Step 1 Divide x into x2.
x
x
(x + 2»x2 - x - 7 x Jx2
Step 2 Multiply and subtract product.
x+
x
(x)(x + 2) = x2 + 2x
2)x'"----2---x---7
x2- x
2
- (x + 2x)
- xL 2x
- 3x
0 - 3x
Step 3 Bring down - 7 and divide x into 3x.
x-3 x + 2)x2 - x- 7
- (x2 + 2x)
- 3x - 7
-3
x) - 3x
Step 4
Multiply and subtract product.
23 15)
(3)(15) = 45
346 30 46 46 - 45 = 1
45
1 remainder
Step 4 Multiply and subtract product. ,----,._x_-_3 (-3)(x+2) = -3x-6 x + 2) x2 - x - 7
- (x2 + 2x)
- 3x - 7
- (- 3x - 6)
o - 1 remainder
Step 5 Check by multiplication.
(23)(15) + 1 = 346
Step 5 Check by multiplication. (x +
2)(x - 3) -1 = x2 - X - 6 - 1 =x2-x-7
Exponents and Polynomials Chapter 5· 145
Check to be sure
the powers are in
descending order
before dividing a
polynomial.
If the coefficient of one power of the variable is zero, then mark its
place with a zero. Be sure to keep terms with the same power
aligned in the same column.
C8pL 125)
Find the quotient of 2p -5
.
4p2 + 10p + 25
2p - 5 ) 8p3 + 0
+ 0 - 125 _(8p3 - 20p2)
+ 20p2 + 0
_(20p2 - 50p)
+ 50p - 125 (50p - 125)
o
0
Check. (4p2 + 10p + 25)(2p - 5) 8p3
+ 20p2 + 50p - 20p2 - 50p - 125
:i2 - 2x
- 8
1.
(x+ 2)
Exercise
A+
Find
:i2 + 8x
15 each quotient.
2.
(x+ 5)
:i2 - 5x
+6
3.
(x- 3)
:i2 -3x
+2
4.
(x - 1)
:i2 +
5x - 50
5.
(x - 5)
:i2 - llx + 28
6.
(x- 4)
7.
146
:i2 +
24
lOx +
(x+ 6)
Chapter 5 Exponents and Polynomials
x2 - 36
8.
(x - 6)
:i2-4
9.
(x+ 2)
10
.
x2 - 100
(x - 10)
8p3 - 125 True
Exercise B Find each quotient. Identify any remainder.
x2 - 7x + 11
x2 - Sx + 15
11.
14.
(x - 2)
(x - 4)
x2 + x-IS
x2 - 2x - 22
12.
15.
(x + 4)
(x + 4)
xl + 9x + 9
13.
(x + 7)
Exercise C Find each quotient.
15m3
-
5m2
- 6m + 2
21.
16.
x3 + x2 - 22x + S
(x - 4)
(3m - 1)
27p3 - S
-Sm2 -D
14m
- 16 your answers to 22.
Exercise
Check
Exercises A, B, and C by
(3p any
- 2) remainder.
17.
multiplying (quotient· divisor) and adding
(2m + 5)
22C - 5['2 + 39
a2 - 16
23.
(2y + 3)
18.
(a - 4)
Sp3 - 125
24. (2p - 5) 3
122C - 3i - 20[, + 5
19.
Estimate: Find the graphed relationship between
x and x2. y
15z4 - 15z + 1
(4y - 1)
20.
3x3 + 7x2 - 7x - 9
25.
(3z3 - 3)
(3x + 1)
Solution: Look at the graph. Is x3 > x2 for all positive values of x? x3 = x2
when x = 0 or 1 . x3 < x2 when 0 < x < 1 . x3 > x2 when x> 1.
Exponents and Polynomials Chapter 5
147
Polynomial Interest
One type of savings account is a simple
savings account. If you invest an amount,
P, called the principal, at an annual
interest rate of r. you will have P + (P' r)
dollars at the end of one year. If you let 5
stand for savings, you can write this as
5 = P + tP» r) or 5 = P (1 + r).
How much savings will you have if you
place $100 for one year in an account that
pays 5 percent interest?
5 = P(1 + r)
5 = $100(1 + .05) = $105
If you leave your savings in the account
for a second year, the new principal is
$105. The amount of savings becomes
5 = [P(1 + r)] (1 + r) or 5 = P(1 + r)2
If a principal, P, is invested at an interest
rate, r, and interest is compounded
annually for tyears, the total savings will
be 5 = P(1 + r)t.
In some savings accounts, interest is
computed (or compounded) more
frequently than once a year. Some
accounts compound interest quarterly
(four times a year); some compound
interest daily!
You place $100 in a savings account. The annual interest rate is 5 percent
and is compounded quarterly. What is your total savings after one year? In
this example, P = $100; r = 5 percent; n = 4, the number of times the interest is
compounded in a year; and t = 1, the number of years.
S = 100(1 + 0.0125)4
S = P(l + !_)nt
n
S = 100(1 + ~)4' 1
4
S = 100(1.0509453) = $105.09
Exercise Find the amount of savings in each instance.
1. $1,000 deposited for one year at 5 percent annual
interest, compounded every six months.
2. $1,000 deposited for one year at 5 percent annual
interest, compounded quarterly.
3. $1,000 deposited for two years at 5 percent
annual interest, compounded quarterly.
4. $1,000 deposited for three years at 8 percent
annual interest, compounded annually.
5. $1,000 deposited for three years at 8 percent
annual interest, compounded quarterly.
150
Chapter 5 Exponents and Polynomials
REVIEW
Chapter 5
Write the letter of the correct answer.
1. Simplify (XS)3.
A x9
C
B
D :x?-
x3
XI8
2. Simplify [(a + b)3]2.
l
A (a + b) 2
C (a + b)1
B (a + b)5
D (a + b)6
3. Find the quotient of ~: .
A4
C 1
B 1J:_
D
4
4.
64
Find the quotient of (4x + 2y)5 -7- (4x + 2yf, 4x + 2y
A (4x + 2y)2
C (4x+ 2y)-2
5
B (4x+ 2y)12
D (4x + 2y) 7
5. What is 4,000,000,000 in scientific notation?
A 4.0. 108
C 409
B 4.0· 109
D 4.0. lOlD
6. What is 3.14(10-4) in standard notation?
A 3\,400
C 0.00314
B 0.000314
D 3.000014
Exponents and Polynomials Chapter 5 151
* O.
REV lEW - continued
Chapter 5
Find the sum, difference, product, or quotient.
Write your answer in scientific notation.
Example: 2(103) + 2(103) Solution: 2(103) + 2(103) = 4(103)
7. 3.1(10-4) + 4.2(10-4)
8. 4.7(10-2) - 3.6(10-2)
9. 1.3(10-6).4.2(10-4). 1.9(102)
10. 8.58(10-4) -i- 4.29(10-4)
11. 0.2825(10-6) + 1.13(102)
Find the sum and the difference for each pair of polynomials.
Example:
Subtract
Solution: Add
x2 + 2y + 2 (x2 - Y + 1)
x2 + 2y + 2
+x2-y+12X2 +
Y+ 3
x2 + 2y + 2
x2 + 2y + 2
(-1 )(x2 - Y + 1)= - x2 + y - 1 3y + 1
12. 3y5 + 2y3 + ly + 3 4y5
+ 5y3 - 7y - 8
13. 2X4- 4X3- 6X2 + 8x + 10 16. 4X4 + 8X3 + 16x2 + 4
4x3 + 2X2 + 16x
14.
b4 + 4b3 - 6b2 + 4b - 1 -3b4
- 3b3 - 5b2 - 4b
152
Chapter 5 Exponents and Polynomials
Find the product.
Example: (x + 2)(x + 2)
Solution: (x + 2)(x + 2) = x(x + 2) + 2(x + 2)
= X2 + 4x
17.2x2(9x2 +
+4
21. (4x + 3)(x - 2)
X - 7)
18.(x + 3)(9x4)
22. (m3 + m2)2
19.(4y2 + 4)(-3y- 3)
23. (m4 + 5)(2m3 + 3m2 + 2)
20. (x + 5)(x + 4)
Find the quotients. Identify any remainder. Use multiplication to
check your answer.
Example: lOx2 + ;OX + 5
Solution: lOx2 + ;OX + 5= 2x2+ 4x + 1
24.
25.
26.
27.
-'.:O:_::2L_r'_-___::_:18:L_y__3:_::6C!..-) 6
(- 30il + 25.x2 - 20x + 10)
(y -2p)2 (p2 -
5
2p) (3.x2 - 6x)
x(x - 2)
(330 + 24x2 + 18x - 12) -;- 3x
28.
29.
(0 - 49x) -;- (x + 7)
30. (5x-l)) 150 - 13x2 - 18x + 4
When you study for chapter tests, practice the
step-by-step formulas and procedures.
Exponents and Polynomials Chapter 5 153