Download Operations with Polynomials

38
Chapter P
P.4
Prerequisites
Operations with Polynomials
What you should learn:
• Write polynomials in standard
form and identify the leading
coefficients and degrees of polynomials
• Add and subtract polynomials
• Multiply polynomials
• Use special products to multiply
polynomials
• Use operations with polynomials
in application problems
Why you should learn it:
Operations with polynomials enable
you to model various aspects of the
physical world, such as the position
of a free-falling object, as shown in
Exercises 163–168 on page 50.
Basic Definitions
An algebraic expression containing only terms of the form axk, where a is any
real number and k is a nonnegative integer, is called a polynomial in one variable or simply a polynomial. Here are some examples of polynomials in one
variable.
3x 8,
x4 3x3 x2 8x 1,
x3 5, and
9x5
In the term axk, a is called the coefficient, and k the degree, of the term. Note
that the degree of the term ax is 1, and the degree of a constant term is 0. Because
a polynomial is an algebraic sum, the coefficients take on the signs between the
terms. For instance,
x3 4x2 3 1x3 4x2 0x 3
has coefficients 1, 4, 0, and 3. Polynomials are usually written in order of
descending powers of the variable. This is referred to as standard form. For
example, the standard form of 3x2 5 x3 2x is
x3 3x2 2x 5.
Standard form
The degree of a polynomial is defined as the degree of the term with the
highest power, and the coefficient of this term is called the leading coefficient of
the polynomial. For instance, the polynomial
3x4 4x2 x 7
is of fourth degree and its leading coefficient is 3.
Definition of Polynomial in x
Let a0, a1, a2, a3, . . . , an be real numbers and let n be a nonnegative
integer. A polynomial in x is an expression of the form
an x n an1x n1 . . . a2 x2 a1x a0
where an 0. The polynomial is of degree n, and the number an is called
the leading coefficient. The number a0 is called the constant term.
The following are not polynomials, for the reasons stated.
• The expression
2x1 5
is not a polynomial because the exponent in 2x1 is negative.
• The expression
x3 3x12
is not a polynomial because the exponent in 3x12 is not a nonnegative
integer.
Section P.4
Operations with Polynomials
39
Example 1 Identifying Leading Coefficients and Degrees
Write the polynomial in standard form and identify the degree and leading coefficient of the polynomial.
(a) 5x2 2x7 4 2x
(b) 16 8x3
(c) 5 x 4 6x3
Solution
Standard Form
Degree
Leading
Coefficient
(a) 5x2 2x7 4 2x
2x7 5x2 2x 4
7
2
(b) 16 8x3
3
8
4
1
Polynomial
8x3
(c) 5 x 4 6x3
16
x 4 6x3 5
Now try Exercise 7.
A polynomial with only one term is a monomial. Polynomials with two
unlike terms are binomials, and those with three unlike terms are trinomials.
Here are some examples.
Monomial: 5x3
Binomial: 4x 3
Trinomial: 2x2 3x 7
The prefix mono means one, the prefix bi means two, and the prefix tri means
three.
Example 2 Evaluating a Polynomial
Find the value of x3 5x2 6x 3 when x 4.
Solution
When x 4, the value of x3 5x2 6x 3 is
x3 5x2 6x 3 43 542 64 3
Substitute 4 for x.
64 80 24 3
Evaluate terms.
5
Simplify.
Now try Exercise 27.
Adding and Subtracting Polynomials
To add two polynomials, simply combine like terms. This can be done in either a
horizontal or a vertical format, as shown in Examples 3 and 4.
Example 3 Adding Polynomials Horizontally
Use a horizontal format to add 2x3 x2 5 and x2 x 6.
Solution
2x3 x2 5 x2 x 6
2x3 x2 x2 x 5 6
2x3
2x2
x1
Now try Exercise 31.
Write original polynomials.
Group like terms.
Combine like terms.
40
Chapter P
Prerequisites
To use a vertical format to add polynomials, align the terms of the polynomials by their degrees, as shown in the following example.
Example 4 Using a Vertical Format to Add Polynomials
Use a vertical format to add 5x3 2x2 x 7, 3x2 4x 7, and
x3 4x2 8.
Solution
5x3 2x2 x 7
3x2 4x 7
x3
4x3
4x2
9x2
8
5x 6
Now try Exercise 33.
To subtract one polynomial from another, add the opposite. You can do this
by changing the sign of each term of the polynomial that is being subtracted and
then adding the resulting like terms.
Example 5 Subtracting Polynomials Horizontally
Use a horizontal format to subtract x3 2x2 x 4 from 3x3 5x2 3.
Solution
Write original
polynomials.
3x3 5x2 3 x3 2x2 x 4
3x3 5x2 3 x3 2x2 x 4
3x3 x3 5x2 2x2 x 3 4
2x3 7x2 x 7
Add the opposite.
Group like terms.
Combine like terms.
Now try Exercise 39.
Study Tip
The common error illustrated to
the right is forgetting to change
two of the signs in the polynomial that is being subtracted.
When subtracting polynomials,
remember to add the opposite of
every term of the subtracted
polynomial.
Be especially careful to get the correct signs when you are subtracting one
polynomial from another. One of the most common mistakes in algebra is to forget to change signs correctly when subtracting one expression from another. Here
is an example.
Wrong sign
x2 2x 3 x2 2x 2 x2 2x 3 x2 2x 2
Common
error
Wrong sign
Example 6 Using a Vertical Format to Subtract Polynomials
Use a vertical format to subtract 3x4 2x3 3x 4 from
4x4 2x3 5x2 x 8.
Solution
4x4 2x3 5x2 x 8
3x4 2x3
3x 4
4x4 2x3 5x2 x 8
3x4 2x3
x4
Now try Exercise 45.
3x 4
5x2
4x 12
Section P.4
Operations with Polynomials
41
Multiplying Polynomials
The simplest type of polynomial multiplication involves a monomial multiplier.
The product is obtained by direct application of the Distributive Property. For
instance, to multiply the monomial 3x by the polynomial 2x2 5x 3, multiply each term of the polynomial by 3x.
3x2x2 5x 3 3x2x2 3x5x 3x3
6x3 15x2 9x
Example 7 Finding Products with Monomial Multipliers
Multiply the polynomial by the monomial.
(a) 2x 73x
(b) 4x22x3 3x 1
Solution
(a) 2x 73x 2x3x 73x
6x2
Distributive Property
21x
Properties of exponents
(b) 4x22x3 3x 1 4x22x3 4x23x 4x21
8x5 12x3 4x2
Distributive Property
Properties of exponents
Now try Exercise 71.
To multiply two binomials, you can use both (left and right) forms of the
Distributive Property. For example, if you treat the binomial 2x 7 as a single
quantity, you can multiply 3x 2 by 2x 7 as follows.
3x 22x 7 3x2x 7 22x 7
3x2x 3x7 22x 27
6x2 21x 4x 14
Outer
First
Product of
First terms
3x 22x 7
Inner
Last
FOIL Diagram
Product of
Outer terms
Product of
Inner terms
Product of
Last terms
6x2 17x 14
The four products in the boxes above suggest that you can put the product of two
binomials in the FOIL form in just one step. This is called the FOIL Method.
Note that the words first, outer, inner, and last refer to the positions of the terms
in the original product (see diagram at the left).
Example 8 Multiplying Binomials (Distributive Property)
Use the Distributive Property to multiply x 2 by x 3.
Solution
x 2x 3 xx 3 2x 3
x2 3x 2x 6
x2 x 6
Now try Exercise 77.
Distributive Property
Distributive Property
Combine like terms.
42
Chapter P
Prerequisites
Example 9 Multiplying Binomials (FOIL Method)
Use the FOIL method to multiply the binomials.
(a) x 3x 9
(b) 3x 42x 1
Solution
F
O
I
L
(a) x 3x 9 x2 9x 3x 27 x2 12x 27
F
O
I
L
(b) 3x 42x 1 6x2 3x 8x 4 6x2 11x 4
Now try Exercise 81.
To multiply two polynomials that have three or more terms, you can use the
same basic principle that you use when multiplying monomials and binomials.
That is, each term of one polynomial must be multiplied by each term of the other
polynomial. This can be done using either a horizontal or a vertical format.
Example 10 Multiplying Polynomials (Horizontal Format)
4x2 3x 12x 5
4x22x 5 3x2x 5 12x 5
8x3 20x2 6x2 15x 2x 5
8x3 26x2 13x 5
Distributive Property
Distributive Property
Combine like terms.
Now try Exercise 97.
When multiplying two polynomials, it is best to write each in standard form
before using either the horizontal or vertical format. This is illustrated in the next
example.
Example 11 Multiplying Polynomials (Vertical Format)
Write the polynomials in standard form and use a vertical format to multiply.
4x2 x 25 3x x2
Solution
With a vertical format, line up like terms in the same vertical columns, much as
you align digits in whole-number multiplication.
4x2
x2
4
3
4x x 2x2
12x3 3x2
20x2
4x 4 11x3 25x2
x 2
3x 5
Write in standard form.
Write in standard form.
x24x2 x 2
6x
5x 10
x 10
Now try Exercise 101.
3x4x2 x 2
54x2 x 2
Section P.4
Operations with Polynomials
43
Polynomials are often written with exponents. As shown in the next
example, the properties of algebra are used to simplify these expressions.
EXPLORATION
Use the FOIL Method to find the
product of
x ax a
Example 12 Multiplying Polynomials
Expand x 43.
where a is a constant.
What do you notice about the
number of terms in your product?
What degree are the terms in your
product?
Solution
x 43 x 4x 4x 4
x 4x 4x 4
Write each factor.
Associative Property of
Multiplication
x2 4x 4x 16x 4
Multiply x 4x 4.
Combine like terms.
x2
8x 16x 4
x2x 4 8xx 4 16x 4
Distributive Property
Distributive Property
x3
4x2
x3
12x2
8x2
32x 16x 64
48x 64
Combine like terms.
Now try Exercise 129.
Example 13 An Area Model for Multiplying Polynomials
Show that x 22x 1 2x2 5x 2.
Solution
An appropriate area model to demonstrate the multiplication of two binomials
would be A lw, the area formula for a rectangle. Think of a rectangle whose
sides are x 2 and 2x 1. The area of this rectangle is
x 22x 1.
x
x
1
1
x+2
x
Another way to find the area is to add the areas of the rectangular parts, as shown
in Figure P.11. There are two squares whose sides are x, five rectangles whose
sides are x and 1, and two squares whose sides are 1. The total area of these nine
rectangles is
2x2 5x 2.
2x + 1
Figure P.11
Area widthlength
Area sum of rectangular areas
Because each method must produce the same area, you can conclude that
x 22x 1 2x2 5x 2.
Now try Exercise 155.
Special Products
Some binomial products have special forms that occur frequently in algebra.
For instance, the product x 3x 3 is called the product of the sum and
difference of two terms. With such products, the two middle terms subtract out,
as follows.
x 3x 3 x2 3x 3x 9
x2 9
Sum and difference of two terms
Product has no middle term.
44
Chapter P
Prerequisites
Another common type of product is the square of a binomial. With this type
of product, the middle term is always twice the product of the terms in the
binomial.
2x 52 2x 52x 5
4x2 10x 10x 25
4x2 20x 25
Square of a binomial
Outer and inner terms are equal.
Middle term is twice the product
of the terms in the binomial.
Special Products
Let u and v be real numbers, variables, or algebraic expressions. Then the
following formulas are true.
Sum and Difference of Same Terms
Example
u vu v u v
3x 43x 4 3x2 42
9x2 16
Square of a Binomial
Example
2
u v 2
u2
2uv 2
4x 92 4x2 24x9 92
16x2 72x 81
v2
u v2 u2 2uv v2
a
a
b
a2
ab
x 62 x2 2x6 62
x2 12x 36
The square of a binomial can also be demonstrated geometrically. Consider
a square, each of whose sides are of length a b. (See Figure P.12). The total
area includes one square of area a2, two rectangles of area ab each, and one
square of area b2. So, the total area is a2 2ab b2.
a+b
Example 14 Finding Special Products
b
b2
ab
a+b
Figure P.12
Multiply the polynomials.
(a) 3x 23x 2
(b) 2x 72
(c) a 2 b2
Solution
(a) 3x 23x 2 3x2 22
Sum and difference of
same terms
9x2 4
(b) 2x 7 2x 22x7 2
2
Simplify.
72
Square of a binomial
4x2 28x 49
Simplify.
(c) a 2 b a 2 2a 2b b
2
2
2
a2 4a 4 2ab 4b b2
Now try Exercise 107.
Square of a binomial
Simplify.
Section P.4
Operations with Polynomials
45
Applications
There are many applications that require the evaluation of polynomials. One
commonly used second-degree polynomial is called a position polynomial. This
polynomial has the form
16t2 v0 t s0
Position polynomial
where t is the time, measured in seconds. The value of this polynomial gives the
height (in feet) of a free-falling object above the ground, assuming no air resistance. The coefficient of t, v0, is called the initial velocity of the object, and the
constant term, s0, is called the initial height of the object. If the initial velocity is
positive, the object was projected upward (at t 0), if the initial velocity is
negative, the object was projected downward, and if the initial velocity is zero,
the object was dropped.
Example 15 Finding the Height of a Free-Falling Object
t=0
t=1
An object is thrown downward from the top of a 200-foot building. The initial
velocity is 10 feet per second. Use the position polynomial
16t2 10t 200
to find the height of the object when t 1, t 2, and t 3 (see Figure P.13).
200 ft
t=2
Solution
When t 1, the height of the object is
Height 1612 101 200
t=3
16 10 200
174 feet.
Figure P.13
When t 2, the height of the object is
Height 1622 102 200
64 20 200
116 feet.
When t 3, the height of the object is
Height 1632 103 200
144 30 200
26 feet.
Now try Exercise 167.
In Example 15, the initial velocity is 10 feet per second. The value is negative
because the object was thrown downward. If it had been thrown upward, the
initial velocity would have been positive. If it had been dropped, the initial
velocity would have been zero.
Use your calculator to determine the height of the object in Example 15 when
t 3.2368. What can you conclude?
46
Chapter P
Prerequisites
Example 16 Using Polynomial Models
The numbers of vehicles (in thousands) fueled by compressed natural gas G and
by electricity E in the United States from 1995 to 2003 can be modeled by
G 0.079t2 8.95t 3.2, 5 ≤ t ≤ 13
E 1.090t2 14.73t 51.6, 5 ≤ t ≤ 13
Vehicles fueled by natural gas
Vehicles fueled by electricity
where t represents the year, with t 5 corresponding to 1995. Find a model that
represents the total numbers T of vehicles fueled by compressed natural gas and
by electricity from 1995 to 2003. Then estimate the total number T of vehicles
fueled by compressed natural gas and by electricity in 2002. (Source: Science
Applications International Corporation and Energy Information Administration)
Solution
The sum of the two polynomial models is as follows.
G E 0.079t2 8.95t 3.2 1.090t2 14.73t 51.6
1.169t2 5.78t 54.8
So, the polynomial that models the total numbers of vehicles fueled by compressed natural gas and by electricity is
TGE
1.169t2 5.78t 54.8
Using this model, and substituting t 12, you can estimate the total number of
vehicles fueled by compressed natural gas and by electricity in 2002 to be
T 1.169122 5.7812 54.8
153.776 thousand vehicles.
Now try Exercise 169.
Example 17 Geometry: Finding the Area of a Shaded Region
Find an expression for the area of the shaded portion in Figure P.14.
2x + 5
x+3
x−3
x+1
Solution
First find the area of the large rectangle A1 and the area of the small rectangle A2.
A1 2x 5x 1 and
Figure P.14
A2 x 3x 3
Then to find the area A of the shaded portion, subtract A2 from A1.
A A1 A2
Write formula.
2x 5x 1 x 3x 3
Substitute.
2x2 7x 5 x2 9
Use FOIL Method and
special product formula.
2x2 7x 5 x2 9
Distributive Property.
x2 7x 14
Combine like terms.
Now try Exercise 149.
Section P.4
P.4
Operations with Polynomials
47
Exercises
VOCABULARY CHECK: Fill in the blanks.
1. The expression an x n an1 x n1 . . . a2 x2 a1x a0an 0 is called a ________.
2. The ________ of a polynomial is the degree of the term with the highest power, and the coefficient of
this term is the ________ of the polynomial.
3. A polynomial with one term is called a ________, while a polynomial with two unlike terms is called
a ________, and a polynomial with three unlike terms is called a ________.
4. The letters in “FOIL” stand for the following. F ________ O ________ I ________ L ________
5. The product u vu v u2 v2 is called the ________ and ________ of ________ terms.
6. The product u v2 u2 2uv v2 is called the ________ of a ________.
7. The expression 16t2 v0t s0 is called the ________, and v0 is the initial ________ and s0 is the initial ________.
In Exercises 1–12, write the polynomial in standard form,
and find its degree and leading coefficient.
1. 10x 4
2.
3. 5 4. 3x3 2x2 3
3y4
3x2
8
5. 8z 16z2
6. 35t 16t2
7. 6t 4t5 t2 3
8. 10 3x2 15x5 7x
9. x 5 x3 5x2
11. x
10. 16 z2 8z 4z3
12. 4
29. x4 4x3 16x 16
(a) x 1
(b) x 52
30. 3t 4 4t3
(a) t 1
(b) t 23
In Exercises 31–34, perform the addition using a horizontal
format.
31. 2x2 3 5x2 6
32. 3x3 2x 8 3x 5
33. x2 3x 8 2x2 4x 3x2
34. 5y 6 4y2 6y 3 9 2y 11y2
In Exercises 13–18, determine whether the polynomial is a
monomial, binomial, or trinomial.
13. 12 5y2
14. t3
15. x3 2x2 4
16. 2u7 9u3
17. 1.3x2
18. 2 x4 4z2
In Exercises 35–38, perform the addition using a vertical
format.
35. 5x2 3x 4 3x2 4
In Exercises 19–26, give an example of a polynomial in one
variable satisfying the conditions. (There are many correct
answers.)
19. A monomial of degree 3
20. A trinomial of degree 3
36. 4x3 2x2 8x 4x2 x 6
37. 2b 3 b2 2b 7 b2
38. v2 v 3 4v 1 2v2 3v
In Exercises 39–42, perform the subtraction using a
horizontal format.
39. 3x2 2x 1 2x2 x 1
21. A trinomial of degree 4 and leading coefficient 2
22. A binomial of degree 2 and leading coefficient 8
23. A monomial of degree 1 and leading coefficient 7
24. A binomial of degree 5 and leading coefficient 3
40. 5y4 2 3y4 2
41. 10x3 15 6x3 x 11
42. y2 3y4 y4 y2
25. A monomial of degree 0
In Exercises 43–46, perform the subtraction using a vertical
format.
26. A monomial of degree 2 and leading coefficient 9
43. x2 x 3 x 2
In Exercises 27–30, evaluate the polynomial for each
specified value of the variable.
27. x3 12x
(a) x 2
(b) x 0
28. 14x4 2x2
(a) x 2
(b) x 2
44. 3z2 z z3 2z2 z
45. 2x3 15x 25 2x3 13x 12
46. 0.2t 4 5t 2 t 4 0.3t 2 1.4
48
Chapter P
Prerequisites
In Exercises 47–68, perform the indicated operation(s).
91. 3x5x5x 2
47. 92. 4t3tt2 1
2
48. 20s 12s 32 15s2 6s
2
2
49. 4x 5x 6 2x 4x 5
3
2
3
50. 13x 9x 4x 5 5x 7x 3
51. 10x2 11 7x3 12x2 15
52. 15y4 18y 18 11y4 8y 8
53. 5s 6s 30s 8
54. 3x2 23x 9 x2
55. 8x3 4x2 3x x3 4x2 5 x 5
56. 5y2 2y y2 y 3y2 6y 2
57. 52x3 1 3x3 12x2 4x 2 3x2 2x 1
58. 2 y2 3y 9 34y 4 5y2 2y 3
59. 2t2 12 5t2 5 6t2 5
60. 10v 2 8v 1 3v 9
61. 2z2 z 11 3z2 4z 5 22z2 5z 10
62. 73t4 2t2 t 5t4 9t2 4t 38t2 5t
63. 25x3 13x 4x3 9x2 3x 3x3 2x2 6x 5
64. 5t3 2t2 t 8 3t3 t2 4t 2 42t2 3t 1 t3 1
2
2
65. 8.04x 9.37x 5.62x2
66. 11.98y3 4.63y3 6.79y3
67. 4.098a2 6.349a 11.246a2 9.342a
68. 27.433k2 19.018k 14.61k2 3.814k
3x2
8 7 5x2
In Exercises 69–96, perform the multiplication and simplify.
69. 2a28a
70. 6n3n2
71. 2y5 y
72. 5z2z 7
73. 4x32x2 3x 5
74. 3y23y2 7y 3
75. 2x25 3x2 7x3
76. 3a211a 3
77. x 7x 4
78. y 2 y 3
79. 5 x3 x
80. 2 y4 y
81. 2t 1t 8
82. 3z 52z 7
83. 3a4 52a4 7
84. 8b5 13b5 2
85. 2x y3x 2y
86. 2x y3x 2y
87. 5x2 3yx2 y
88. a3 2b54a3 3b5
89. 4y 13 12y 9
90. 5t 34 2t 16
93. 5aa 2 3a2a 3
94. 4x2x 1 9xx 3
95. 2t 1t 1 32t 5
96. 58y 3 2y 1y 7
In Exercises 97–100, perform the multiplication using a
horizontal format.
97. x3 3x 2x 2
98. t 3t2 5t 1
99. u 52u2 3u 4
100. x 1x2 4x 6
In Exercises 101–104, perform the multiplication using a
vertical format.
101. 7x2 14x 9x 3
102. 4x4 6x2 92x2 3
103. x2 2x 12x 1
104. 2s2 5s 63s 4
In Exercises 105–138, perform the multiplication.
105. x 4x 4
106. y 7y 7
107. a 6ca 6c
108. 8n m8n m
109. 2t 92t 9
110. 5z 15z 1
111. 2x 1
4
2x 1
4
112.
23 x 723 x 7
113. 0.2t 0.50.2t 0.5
114. 4a 0.1b4a 0.1b
115. x3 4x3 4
116. a5 3a5 3
117. x 52
118. x 92
119. 5x 2
120. 3x 82
121. 2a 3b2
122. 4x 5y2
123. 2x4 32
124. y7 4z2
2
125. x2 4x2 2x 4
126. 2x2 32x2 2x 3
127. t2 5t 12t2 5
128. 2z2 3z 73z 4
129. a 53
130. y 23
131. 2x 33
132. 3y 43
133. a2 9a 5a2 a 3
134. t2 2t 72t2 8t 3
135. x 2 y2
136. x 4 y2
137. 2z y 12
138. u v 32
Section P.4
In Exercises 139–142, perform the indicated operations and
simplify.
139. x 3x 3 x2 8x 2
Operations with Polynomials
49
Geometric Modeling
In Exercises 153–156, (a) perform
the multiplication algebraically and (b) use a geometric
area model to verify your solution to part (a).
153. xx 3
140. k 8k 8 k2 k 3
154. 2yy 1
141. t 32 t 32
155. t 3t 2
142. a 62 a 62
156. 2z 5z 1
Geometry
In Exercises 143–146, write an expression for
the perimeter or circumference of the figure.
143.
a
144.
x2
x
x
a+3
a+3
Geometric Modeling In Exercises 157 and 158, use the
area model to write two different expressions for the total
area. Then equate the two expressions and name the algebraic property that is illustrated.
157.
x
x2
a
x
a
x+b
145.
146.
3y
2y
b
2x − 4
y+4
x+a
x
158.
4y − 5
a
x
Geometry
In Exercises 147–152, write an expression for
the area of the shaded portion of the figure.
147.
148.
3x + 1
x+a
a
2x − 5
y+2
x+a
149.
159. Geometry The length of a rectangle is 112 times its
width w. Write expressions for (a) the perimeter
and (b) the area of the rectangle.
3x + 10
160. Geometry The base of a triangle is 3x and its
height is x 5. Write an expression for the area A
of the triangle.
x
3x
x +4
150.
151.
4x + 2
5t 2
x−1
x−1
4x + 2
6t
7t 4
152.
1.6x
0.8x
x
2x
6t
161. Compound Interest After 2 years, an investment
of $1000 compounded annually at an interest rate
of r will yield an amount 10001 r2. Find this
product.
162. Compound Interest After 2 years, an investment
of $1000 compounded annually at an interest rate
of 3.5% will yield an amount 10001 0.0352.
Find this product.
Chapter P
Prerequisites
Free-Falling Object
In Exercises 163–166, use the position polynomial to determine whether the free-falling object
was dropped, thrown upward, or thrown downward. Then
determine the height of the object at time t 0.
163. 16t2 100
164. 16t2 50t
165. 16t2 24t 50
166. 16t2 32t 300
167. Free-Falling Object An object is thrown upward
from the top of a 200-foot building (see figure). The
initial velocity is 40 feet per second. Use the position polynomial 16t2 40t 200 to find the
height of the object when t 1, t 2, and t 3.
(b) During the given time period, the per capita
consumption of beverage milks was decreasing
and the per capita consumption of bottled water
was increasing (see figure). Was the combined
per capita consumption of both beverage milks
and bottled water increasing or decreasing over
the given time period?
y
Per capita consumption
(in gallons)
50
30
Beverage milks
Bottled water
25
20
15
10
t
5
200 ft
250 ft
6
7
8
9
10
11
12
13
Year (5 ↔1995)
Figure for 169
Synthesis
Figure for 167
Figure for 168
168. Free-Falling Object An object is thrown downward from the top of a 250-foot building (see
figure). The initial velocity is 25 feet per second.
Use the position polynomial 16t2 25t 250
to find the height of the object when t 1, t 2,
and t 3.
169. Beverage Consumption The per capita consumption of all beverage milks M and bottled water W in
the United States from 1995 to 2003 can be approximated by the following two polynomial models.
M 0.0009t2 0.288t 25.44 Beverage milks
W 0.0544t2 0.268t 9.40
Bottled water
In these models, the per capita consumption is
given in gallons and t5 ≤ t ≤ 13 represents the
year, with t 5 corresponding to 1995. (Source:
USDA/Economic Research Service)
(a) Find a polynomial model that represents the per
capita consumption of both beverage milks and
bottled water during the given time period. Use
this model to find the per capita consumption of
beverage milks and bottled water in 1999 and
2003.
170. Writing Explain why x y2 is not equal to
x2 y2.
171. Think About It Is every trinomial a seconddegree polynomial? Explain.
172. Think About It Can two third-degree polynomials
be added to produce a second-degree polynomial?
If so, give an example.
173. Perform the multiplications.
(a) x 1x 1
(b) x 1x2 x 1
(c) x 1x3 x2 x 1
From the pattern formed by these products, can
you predict the result of x 1x4 x3 x2 x 1?
174. Writing Explain why x2 3x23 is not a
polynomial.