6.1
Introduction to Polynomials and
Polynomial Functions
6.1
OBJECTIVES
1. Identify like terms
2. Find the degree of a polynomial
3. Find an ordered pair associated with a given
polynomial function
In Chapter 4, we looked at a class of functions called linear functions. In this section, we
examine polynomial functions. We begin by defining some important words.
Definitions: Term
A term is a number or the product of a number and one or more variables,
raised to a power.
Example 1
Identifying Terms
Which of the following are terms?
2x2 3x
5x3
4xy
5x and 4xy are terms. 2x2 3x is not a term; it is the sum of two terms.
3
CHECK YOURSELF 1
Which of the following are terms?
(b) 4x32y
(a) 5xy
(c) 2x3y2
(d) x7
If terms contain exactly the same variables raised to the same powers, they are called
like terms. Examples include 6s and 7s, 4x2 and 9x2, and 7xy2z3 and 10xy2z3. The following
are not like terms
Different variables
6s and 7t
Different exponents
4x2 and 9x3
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Different exponents
2 3 3
2 3
7x y z and 10xy z
Example 2
Identifying Like Terms
For each of the following pairs of terms, decide whether they are like terms.
(a) 5x3 and 5x2
Not like terms—different exponents
379
380
CHAPTER 6
POLYNOMIALS AND POLYNOMIAL FUNCTIONS
(b) 3xy and 2xy
Like terms
(c) 4xy2z and 9xy2z
Like terms
(d) 2xy2 and 7x2y
Not like terms—different exponents
(e) 3x2y and 2x2z
Not like terms—different variables
CHECK YOURSELF 2
For each of the following pairs of terms, decide whether they are like terms.
NOTE The prefix “mono”
means one.
1 3
xy
2
(a) 2ab2c and 3ab2d
(b) 4xy3 and
(c) 5x2y3z 2 and 7x3y2z 2
(d) 3x and 4xy
A polynomial consists of one or more terms in which the only allowable exponents are
the whole numbers, 0, 1, 2, 3, . . . and so on. The terms are connected by addition or subtraction signs.
Certain polynomials are given special names according to the number of terms that they
have.
A polynomial with one term is called a monomial. For example,
7x3
2x2y3
4xy
12
are all polynomials. But,
8x4
9x3y4
6x2y7
10xy
the numerical coefficients are 8, 9, 6, and 10.
Example 3
Classifying Polynomials
Which of the following are polynomials? Classify the polynomials as monomial, binomial,
or trinomial.
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NOTE The prefix “bi” means
two and “tri” means three.
9
5 1x
x
are not polynomials because the variable in the first term is in the denominator and the variable in the second term is under the radical (we will see in Chapter 8 that the exponent in
1
this case is ).
2
A polynomial with two terms is a binomial. A polynomial with three terms is called a
trinomial.
The numerical factor in a term is called the numerical coefficient, or more simply the
coefficient, of that term. For example, in the terms
INTRODUCTION TO POLYNOMIALS AND POLYNOMIAL FUNCTIONS
SECTION 6.1
381
(a) 5x2y
Monomial
(b) 3m 5n
Binomial
(c) 4a3 3a 2
Trinomial
NOTE Remember that
1
x 1
x
2
x
2
5y2 is not a polynomial because the variable x is in the denominator.
x
(d) 5y2 CHECK YOURSELF 3
Which of the following are polynomials? Classify the polynomials as monomial,
binomial, or trinomial.
(a) 5x2 6x
(b) 8x5
(c) 5x3 3xy 7y2
(d) 9x 3
x
It is also useful to classify polynomials by their degree.
Definitions: Degree of Monomials
The degree of a monomial is the sum of the exponents of the variable factors.
Example 4
Determining the Degree of a Monomial
(a) 5x2 has degree 2.
(b) 7n5 has degree 5.
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(c) 3a2b4 has degree 6. (The sum of the powers, 2 and 4, is 6.)
(d) 9 has degree 0 (because 9 9 1 9x0).
CHECK YOURSELF 4
Give the degree of each monomial.
(a) 4x2
(c) 8p2s
(b) 7x3y2
(d) 5
CHAPTER 6
POLYNOMIALS AND POLYNOMIAL FUNCTIONS
Definitions: Degree of Polynomials
The degree of a polynomial is that of the term with the highest degree.
Example 5
Determining the Degree of a Polynomial
(a) 7x3 5x2 5 has degree 3.
(b) 5y7 3y2 5y 7 has degree 7.
(c) 4a2b3 5abc2 has degree 5 because the sum of the variable powers in the term with
highest degree (4a2b3) is 5.
Polynomials such as those in Examples 5(a) and 5(b) are called polynomials in one
variable, and they are usually written in descending form so that the power of the variable
decreases from left to right. In that case, the coefficient of the first term is called the leading coefficient.
CHECK YOURSELF 5
Give the degree of each polynomial. For those polynomials in one variable, write in
descending form and give the leading coefficient.
(a) 7x4 5xy 2
(b) 5
(c) 4x2 7x3 8x 5
A polynomial function is a function in which the expression on the right-hand side is a
polynomial expression. For example,
f(x) 3x3 2x2 x 5
g(x) 2x3 5x 1
P(x) 2x7 2x6 x3 2x4 5x2 6
are all polynomial functions. Because they are functions, every x value determines a unique
ordered pair.
Example 6
Finding Ordered Pairs
Given f(x) 3x3 2x2 x 5 and g(x) 2x3 5x 1, find the following ordered
pairs.
(a) (0, f(0))
To find f(0), we substitute 0 for x in the function f(x) 3x3 2x2 x 5.
f(0) 3(0)3 2(0)2 (0) 5
0005
5
Therefore, (0, f(0)) (0, 5).
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382
INTRODUCTION TO POLYNOMIALS AND POLYNOMIAL FUNCTIONS
SECTION 6.1
383
(b) (2, f(2))
f(2) 3(2)3 2(2)2 (2) 5
24 8 2 5
35
Therefore, (2, f(2)) (2, 35).
(c) (2, f(2))
f(2) 3(2)3 2(2)2 (2) 5
24 8 2 5
9
Therefore, (2, f(2)) (2, 9).
(d) (0, g(0))
g(0) 2(0)3 5(0) 1
1
Therefore, (0, g(0)) (0, 1).
(e) (2, g(2))
g(2) 2(2)3 5(2) 1
16 10 1
5
Therefore, (2, g(2)) (2, 5).
(f) (2, g(2))
g(2) 2(2)3 5(2) 1
16 10 1
7
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Therefore, (2, g(2)) (2, 7).
CHECK YOURSELF 6
Given f(x) x 3 7x 1 and g(x) x 3 2x 2 3x 4, find the following ordered
pairs.
(a) (0, f(0))
(c) (2, f(2))
(e) (2, g(2))
(b) (2, f(2))
(d) (0, g(0))
(f) (2, g(2))
CHAPTER 6
POLYNOMIALS AND POLYNOMIAL FUNCTIONS
CHECK YOURSELF ANSWERS
(a) Term; (b) not a term; (c) term; (d) term
(a) Not like terms; (b) like terms; (c) not like terms; (d) not like terms
(a) Binomial; (b) monomial; (c) trinomial; (d) not a polynomial
(a) 2; (b) 5; (c) 3; (d) 0
(a) Degree is 4; (b) degree is 0; (c) degree is 3, 7x3 4x2 8x 5, leading
coefficient is 7
6. (a) (0, 1); (b) (2, 5); (c) (2, 7); (d) (0, 4); (e) (2, 6); (f) (2, 2)
1.
2.
3.
4.
5.
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384
Name
6.1
Exercises
Section
Date
Which of the following expressions are polynomials?
1. 7x3
2. 5x3 3
x
ANSWERS
1.
2.
3. 4x4y2 3x3y
4. 7
3.
4.
5. 7
6. 4x3 x
5.
6.
7.
3x
x2
8. 5a2 2a 7
7.
8.
For each of the following polynomials, list the terms and the coefficients.
9. 2x2 3x
10. 5x3 x
9.
10.
11.
11. 4x3 3x 2
12.
12. 7x2
13.
For each of the following pairs of terms, decide whether they are like terms.
14.
13. 3xy2c and 2xy2c
15.
14. 5xy3 and 4xy3
16.
15. 6xy and 2x2y2
16. 3x3y4 and 2x4y3
17.
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18.
Classify each of the following as a monomial, binomial, or trinomial where possible.
19.
17. 7x3 3x2
20.
18. 4x7
21.
19. 7y2 4y 5
20. 2x2 3xy y2
21. 2x4 3x2 5x 2
22. x4 22.
5
7
x
385
ANSWERS
23.
24. 4x4 2x2 5x 7
23. 6y8
24.
25.
25. x5 26.
3
x2
26. 4x2 9
27.
28.
Arrange in descending-exponent form if necessary, and give the degree of each
polynomial.
29.
27. 4x5 3x2
28. 5x2 3x3 4
29. 7x7 5x9 4x3
30. 2 x
31. 4x
32. x17 3x4
33. 5x2 3x5 x6 7
34. 5
30.
31.
32.
33.
34.
35.
36.
37.
Find the values of each of the following polynomials for the given values of the variable.
38.
35. 6x 1; x 1 and x 1
36. 5x 5; x 2 and x 2
37. x3 2x; x 2 and x 2
38. 3x2 7; x 3 and x 3
39. 3x2 4x 2; x 4 and x 4
40. 2x2 5x 1; x 2 and x 2
41. x2 2x 3; x 1 and x 3
42. x2 5x 6; x 3 and x 2
39.
40.
41.
42.
43.
45.
46.
Indicate whether each of the following statements is always true, sometimes true, or
never true.
386
43. A monomial is a polynomial.
44. A binomial is a trinomial.
45. The degree of a trinomial is 3.
46. A trinomial has three terms.
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44.
ANSWERS
47. A polynomial has four or more terms.
48. A binomial must have two coefficients.
47.
48.
49.
49. If x equals 0, the value of a polynomial
50. The coefficient of the leading term in
in x equals 0.
a polynomial is the largest coefficient
of the polynomial.
In exercises 51 to 58, the polynomial functions f(x) and g(x) are given. Find the ordered
pairs (a) (0, f(0)), (b) (2, f(2)), (c) (2, f(2)), (d) (0, g(0)), (e) (2, g(2)), and
(f) (2, g(2))
51. f(x) 2x2 3x 5 and g(x) 4x2 5x 7
50.
51. (a)
(b)
(c)
(d)
(e)
(f)
52. (a)
(b)
(c)
(d)
(e)
(f)
53. (a)
52. f(x) 3x2 7x 9 and g(x) 7x2 8x 9
(b)
(c)
(d)
(e)
(f)
54. (a)
53. f(x) x3 8x2 4x 10 and g(x) 3x3 4x2 5x 3
(b)
(c)
(d)
(e)
(f)
55. (a)
54. f(x) x3 3x2 7x 8 and g(x) 2x3 4x2 9x 2
55. f(x) 4x3 5x2 8x 12 and g(x) x3 5x2 7x 14
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56. f(x) 7x3 5x 9 and g(x) 4x3 3x2 8
57. f(x) 10x3 5x2 3x and g(x) 8x3 4x2 9
(b)
(c)
(d)
(e)
(f)
56. (a)
(b)
(c)
(d)
(e)
(f)
57. (a)
(b)
(c)
(d)
(e)
(f)
58. (a)
58. f(x) 7x2 5 and g(x) 3x3 5x2 6x 9
(b)
(c)
(d)
(e)
(f)
387
ANSWERS
59.
59. Cost of typing. The cost, in dollars, of typing a term paper is given as three times
the number of pages plus 20. Use x as the number of pages to be typed and write a
polynomial to describe this cost. Find the cost of typing a 50-page paper.
60.
61.
60. Manufacturing. The cost, in dollars, of making suits is described as 20 times the
62.
number of suits plus 150. Use x as the number of suits and write a polynomial to
describe this cost. Find the cost of making seven suits.
63.
64.
61. Revenue. The revenue, in dollars, when x pairs of shoes are sold is given by
R(x) 3x2 95. Find the revenue when 12 pairs of shoes are sold.
62. Manufacturing. The cost in dollars of manufacturing x wing nuts is given by
C(x) 0.07x 13.3. Find the cost when 375 wing nuts are made.
Let P(x) 2x3 3x2 5x 5 and Q(x) x2 2x 2. In exercises 63 and 64, find
each of the following.
63. P[Q(1)]
64. Q[P(1)]
1. Polynomial
3. Polynomial
5. Polynomial
7. Not a polynomial
9. 2x2, 3x; 2, 3
11. 4x3, 3x, 2; 4, 3, 2
13. Like terms
15. Not like terms
17. Binomial
19. Trinomial
21. Not classified
23. Monomial
25. Not a polynomial
27. 4x5 3x2; 5
29. 5x9 7x7 4x3; 9
31. 4x; 1
33. x6 3x5 5x2 7; 6
35. 7, 5
37. 4, 4
39. 62, 30
41. 0, 0
43. Always
45. Sometimes
47. Sometimes
49. Sometimes
51. (a) (0, 5); (b) (2, 9); (c) (2, 3); (d) (0, 7); (e) (2, 1); (f) (2, 19)
53. (a) (0, 10); (b) (2, 42); (c) (2, 42); (d) (0, 3); (e) (2, 27); (f) (2, 1)
55. (a) (0, 12); (b) (2, 48); (c) (2, 16); (d) (0, 14); (e) (2, 12); (f) (2, 0)
57. (a) (0, 0); (b) (2, 94); (c) (2, 54); (d) (0, 9); (e) (2, 89); (f) (2, 39)
59. C(x) 3x 20; $170
61. $337
63. 15
388
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Answers