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This is “Introduction to Polynomials”, section 5.2 from the book Beginning Algebra (v. 1.0). For details on it (including licensing), click here.
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5.2 Introduction to Polynomials
Learning Objectives
1.Identify a polynomial and determine its degree.
2.Evaluate a polynomial for given values of the variables.
3.Evaluate a polynomial using function notation.
Definitions
A polynomialAn algebraic expression consisting of terms with real number coefficients and variables with whole number exponents. is a special algebraic expression with terms that consist of real number coefficients and variable factors with whole number exponents.
Polynomials do not have variables in the denominator of any term.
The degree of a termThe exponent of the variable; if there is more than one variable in the term, the degree of the term is the sum their exponents. in a polynomial is defined to be the exponent of the variable, or if there is more than one variable in the term, the degree is the sum of their exponents. Recall that x0=1; any constant term can be written as a product of x0 and itself. Hence the degree of a constant term is 0.
Term
Degree
3x2
2
6x2y
2+1=3
7a2b3
2+3=5
8
0 , since 8=8x0
2x
1, since x=x1
The degree of a polynomialThe largest degree of all of its terms. is the largest degree of all of its terms.
Polynomial
Degree
4x5−3x3+2x−1
5
6x2y−5xy3+7
4 , because 5xy3 has degree 4.
12x+54
1, because x=x1
We classify polynomials by the number of terms and the degree as follows:
Expression
Classification
Degree
5x7
MonomialPolynomial with one term. (one term)
7
8x6−1
BinomialPolynomial with two terms. (two terms)
6
−3x2+x−1
TrinomialPolynomial with three terms. (three terms)
2
5x3−2x2+3x−6
PolynomialAn algebraic expression consisting of terms with real number coefficients and variables with whole number exponents. (many terms)
3
In this text, we will call polynomials with four or more terms simply polynomials.
Example 1: Classify and state the degree: 7x2−4x5−1.
Solution: Here there are three terms. The highest variable exponent is 5. Therefore, this is a trinomial of degree 5.
Answer: Trinomial; degree 5
Example 2: Classify and state the degree: 12a5bc3.
Solution: Since the expression consists of only multiplication, it is one term, a monomial. The variable part can be written as a5b1c3; hence its degree is 5+1+3=9.
Answer: Monomial; degree 9
Example 3: Classify and state the degree: 4x2y−6xy4+5x3y3+4.
Solution: The term 4x2y has degree 3; −6xy4 has degree 5; 5x3y3 has degree 6; and the constant term 4 has degree 0. Therefore, the polynomial has 4 terms with degree 6.
Answer: Polynomial; degree 6
Of particular interest are polynomials with one variableA polynomial where each term has the form anxn, where an is any real number and n is any whole number., where each term is of the form anxn. Here an is any real number and n is any whole number. Such polynomials have the standard form
Typically, we arrange terms of polynomials in descending order based on the degree of each term. The leading coefficientThe coefficient of the term with the largest degree. is the coefficient of the variable with the highest power, in this case, an.
Example 4: Write in standard form: 3x−4x2+5x3+7−2x4.
Solution: Since terms are separated by addition, write the following:
In this form, we can see that the subtraction in the original corresponds to negative coefficients. Because addition is commutative, we can write the terms in descending order based on the degree of each term as follows:
Answer: −2x4+5x3−4x2+3x+7
We can further classify polynomials with one variable by their degree as follows:
Polynomial
Name
5
Constant (degree 0)
2x+1
Linear (degree 1)
3x2+5x−3
Quadratic (degree 2)
x3+x2+x+1
Cubic (degree 3)
7x4+3x3−7x+8
Fourth-
In this text, we call any polynomial of degree n≥4 an nth-
Evaluating Polynomials
Given the values for the variables in a polynomial, we can substitute and simplify using the order of operations.
Example 5: Evaluate: 3x−1, where x=−32.
Solution: First, replace the variable with parentheses and then substitute the given value.
Answer: −11/2
Example 6: Evaluate: 3x2+2x−1, where x=−1.
Solution:
Answer: 0
Example 7: Evaluate: −2a2b+ab2−7, where a=3 and b=−2.
Solution:
Answer: 41
Example 8: The volume of a sphere in cubic units is given by the formula V=43πr3, where r is the radius. Calculate the volume of a sphere with radius r=32 meters.
Solution:
Answer: 92π cubic meters
Try this! Evaluate: x3−x2+4x−2, where x=−3.
Answer: −50
Video Solution
(click to see video)
Polynomial Functions
Polynomial functions with one variable are functions that can be written in the form
where an is any real number and n is any whole number. Some examples of the different classes of polynomial functions are listed below:
Polynomial function
Name
f(x)=5
Constant functionA polynomial function with degree 0. (degree 0)
f(x)=−2x+1
Linear functionA polynomial function with degree 1. (degree 1)
f(x)=5x2+4x−3
Quadratic functionA polynomial function with degree 2. (degree 2)
f(x)=x3−1
Cubic functionA polynomial function with degree 3. (degree 3)
f(x)=4x5+3x4−7
Polynomial function
Since there are no restrictions on the values for x, the domain of any polynomial function consists of all real numbers.
Example 9: Calculate: f(5), given f(x)=−2x2+5x+10.
Solution: Recall that the function notation f(5) indicates we should evaluate the function when x=5. Replace every instance of the variable x with the value 5.
Answer: f(5)=−15
Example 10: Calculate: f(−1), given f(x)=−x3+2x2−4x+1.
Solution: Replace the variable x with −1.
Answer: f(−1)=8
Try this! Given g(x)=x3−2x2−x−4, calculate g(−1).
Answer: g(−1)=−6
Video Solution
(click to see video)
Key Takeaways
•Polynomials are special algebraic expressions where the terms are the products of real numbers and variables with whole number exponents.
•The degree of a polynomial with one variable is the largest exponent of the variable found in any term.
•The terms of a polynomial are typically arranged in descending order based on the degree of each term.
•When evaluating a polynomial, it is a good practice to replace all variables with parentheses and then substitute the appropriate values.
•All polynomials are functions.
Topic Exercises
Part A: Definitions
Classify the given polynomial as linear, quadratic, or cubic.
1. 2x+1
2. x2+7x+2
3. 2−3x2+x
4. 4x
5. x2−x3+x+1
6. 5−10x3
Classify the given polynomial as a monomial, binomial, or trinomial and state the degree.
7. x3−1
8. x2y2
9. x−x5+1
10. x2+3x−1
11. 5ab4
12. 13x−12
13. −5x3+2x+1
14. 8x2−9
15. 4x5−5x3+6x
16. 8x4−x5+2x−3
17. 9x+7
18. x5+x4+x3+x2−x+1
19. 6x−1+5x4−8
20. 4x−3x2+3
21. 7
22. x2
23. 4x2y−3x3y3+xy3
24. a3b2−6ab
25. a3b3
26. x2y−y2x
27. xy−3
28. a5bc2+3a9−5a4b3c
29. −3x10y2z−xy12z+9x13+30
30. 7x0
Write the following polynomials in standard form.
31. 1−6x+7x2
32. x−9x2−8
33. 7−x3+x7−x2+x−5x5
34. a3−a9+6a5−a+3−a4
Part B: Evaluating Polynomials
35. Fill in the following chart:
36. Fill in the following chart:
Evaluate.
37. 2x−3, where x=3
38. x2−3x+5, where x=−2
39. −12x+13, where x=−13
40. −x2+5x−1, where x=−12
41. −2x2+3x−5, where x=0
42. 8x5−27x3+81x−17, where x=0
43. y3−2y+1, where y=−2
44. y4+2y2−32, where y=2
45. a3+2a2+a−3, where a=−3
46. x3−x2, where x=5
47. 34x2−12x+36, where x=−23
48. 58x2−14x+12, where x=4
49. x2y+xy2, where x=2 and y=−3
50. 2a5b−ab4+a2b2, where a=−1 and b=−2
51. a2−b2, where a=5 and b=−6
52. a2−b2, where a=34 and b=−14
53. a3−b3, where a=−2 and b=3
54. a3+b3, where a=5 and b=−5
For each problem, evaluate b2−4ac, given the following values.
55. a=−1, b=2, and c=−1
56. a=2, b=−2, and c=12
57. a=3, b=−5, c=0
58. a=1, b=0, and c=−4
59. a=14, b=−4, and c=2
60. a=1, b=5, and c=6
The volume of a sphere in cubic units is given by the formula V=43πr3, where r is the radius. For each problem, calculate the volume of a sphere given the following radii.
61. r = 3 centimeters
62. r = 1 centimeter
63. r = 1/2 feet
64. r = 3/2 feet
65. r = 0.15 in
66. r = 1.3 inches
The height in feet of a projectile launched vertically from the ground with an initial velocity v0 in feet per second is given by the formula h=−16t2+v0t, where t represents time in seconds. For each problem, calculate the height of the projectile given the following initial velocity and times.
67. v0=64 feet/second, at times t = 0, 1, 2, 3, 4 seconds
68. v0=80 feet/second, at times t = 0, 1, 2, 2.5, 3, 4, 5 seconds
The stopping distance of a car, taking into account an average reaction time, can be estimated with the formula d=0.05v2+1.5, where d is in feet and v is the speed in miles per hour. For each problem, calculate the stopping distance of a car traveling at the given speeds.
69. 20 miles per hour
70. 40 miles per hour
71. 80 miles per hour
72. 100 miles per hour
Part C: Polynomial Functions
Given the linear function f(x)=23x+6, evaluate each of the following.
73. f(−6)
74. f(−3)
75. f(0)
76. f(3)
77. Find x when f(x)=10.
78. Find x when f(x)=−4.
Given the quadratic function f(x)=2x2−3x+5, evaluate each of the following.
79. f(−2)
80. f(−1)
81. f(0)
82. f(2)
Given the cubic function g(x)=x3−x2+x−1, evaluate each of the following.
83. g(−2)
84. g(−1)
85. g(0)
86. g(1)
The height in feet of a projectile launched vertically from the ground with an initial velocity of 128 feet per second is given by the function h(t)=−16t2+128t, where t is in seconds. Calculate and interpret the following.
87. h(0)
88. h(12)
89. h(1)
90. h(3)
91. h(4)
92. h(5)
93. h(7)
94. h(8)
Part D: Discussion Board Topics
95. Find and share some graphs of polynomial functions.
96. Explain how to convert feet per second into miles per hour.
97. Find and share the names of fourth-
Answers
1: Linear
3: Quadratic
5: Cubic
7: Binomial; degree 3
9: Trinomial; degree 5
11: Monomial; degree 5
13: Trinomial; degree 3
15: Trinomial; degree 5
17: Binomial; degree 1
19: Not a polynomial
21: Monomial; degree 0
23: Trinomial; degree 6
25: Monomial; degree 6
27: Binomial; degree 2
29: Polynomial; degree 14
31: 7x2−6x+1
33: x7−5x5−x3−x2+x+7
35:
37: 3
39: 1/2
41: −5
43: −3
45: −15
47: 7/6
49: 6
51: −11
53: −35
55: 0
57: 25
59: 14
61: 36π cubic centimeters
63: π/6 cubic feet
65: 0.014 cubic inches
67:
Time
Height
t = 0 seconds h = 0 feet
t = 1 second h = 48 feet
t = 2 seconds h = 64 feet
t = 3 seconds h = 48 feet
t = 4 seconds h = 0 feet
69: 21.5 feet
71: 321.5 feet
73: 2
75: 6
77: x=6
79: 19
81: 5
83: −15
85: −1
87: The projectile is launched from the ground.
89: The projectile is 112 feet above the ground 1 second after launch.
91: The projectile is 256 feet above the ground 4 seconds after launch.
93: The projectile is 112 feet above the ground 7 seconds after launch.
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