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Topic 8
Review
TOPIC VOCABULARY
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Check Your Understanding
Match each vocabulary term with the description that best fits it.
1. Conjugate Root Theorem
A. determines P (a) by dividing the polynomial by x - a
2. Fundamental Theorem of Algebra
B. the degree equals the number of roots
3. Rational Root Theorem
C. minimizes guessing fraction and integer solutions
4. Remainder Theorem
D. complex numbers as roots come in pairs
8-1 Attributes of Polynomial Functions
Quick Review
Exercises
The standard form of a polynomial function is
P (x) = anx n + an-1x n-1 + c + a1x + a0, where n is a
nonnegative integer and the coefficients are real numbers.
A polynomial function is classified by degree. Its degree is
the highest degree among its monomial term(s). The degree
determines the possible number of turning points in the
graph and the end behavior of the graph.
Write each polynomial function in standard form, classify
it by degree, and determine the end behavior of its graph.
Example
Classify P(x) = -4x2 + x4 by degree. How many possible
turning points are there given the degree of P(x)?
The degree of P(x) is 4, so its graph has either 1 or 3
turning points.
5. y = 12 - x 4
6. y = 2x 2 + 8 - 4x + x 3
7. y = x 2 + 7 - x
8. y = 10 - 3x 3 + 3x 2 + x 4
9. If the volume of a cube can be represented by a
polynomial of degree 9, what is the degree of the
polynomial that represents each side length?
10. A polynomial function P(x) has degree n. If n is even, is
the number of turning points of the graph of P(x) even
or odd? What can you say about the number of turning
points if n is odd?
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8-2 Adding, Subtracting, and Multiplying Polynomials
Quick Review
Exercises
You can add and subtract polynomials by combining
like terms.
Find each sum.
You can multiply polynomials using the Distributive
Property or the FOIL method.
11. (4x2 + 3x - 5) + (2x + 4)
12. (6x3 - x2 + 11x) + (3x3 - 9x + 6)
Find each difference.
Examples
What is the sum of (6x2 + 3x - 8) and (5x2 - 4x)?
Combine like terms.
(6x2 + 3x - 8) + (5x2 - 4x)
= (6x2 + 5x2) + (3x - 4x) - 8
= 11x2 - x - 8
What is the product of (4x + 5) and (2x - 1)?
Use the FOIL method. Then combine like terms.
(4x + 5)(2x - 1) = 8x2 - 4x + 10x - 5
13. (x2 - 8x + 7) - (2x2 - 5x - 6)
14. (12x3 - 7x2 + 4x) - (5x3 - x2 - 2)
Find each product.
15. (2 - 2x2)(x + 4)
16. (2x + 1)(3x2 - x + 4)
17. (x + 5)(2x - 1)(x - 2)
= 8x2 + 6x - 5
8-3 Polynomials, Linear Factors, and Zeros
Quick Review
Exercises
For any real number a and polynomial P (x), if x - a is a
factor of P (x), then a is:
Write a polynomial function with the given zeros.
r a zero of y = P (x)
r a root (or solution) of P (x) = 0, and
r an x-intercept of the graph of y = P (x).
If a is a multiple zero, its multiplicity is the same as the
number of times x - a appears as a factor.
18. x = -1, -1, 6
19. x = -1, 0, 2
20. x = 1, 2, 3
21. x = -2, 1, 4
Find the zeros of each function. State the multiplicity of
any multiple zeros.
A turning point is a relative maximum or relative
minimum of a polynomial function.
22. y = 3x(x + 2) 3
23. y = x 4 - 8x 2 + 16
24. y = 4x 3 - 2x 2 - 2x
25. y = (x - 5)(x + 2) 2
Example
Use a graphing calculator to find the relative maximum,
relative minimum, and zeros of each function.
Find the zeros for y = 3x 3 - 6x 2 + 3x and state the
multiplicity of any multiple zeros.
26. f (x) = x 4 - 5x 3 + 5x 2 - 3
y = 3x(x2 - 2x + 1)
Factor out the GCF, 3x.
27. f (x) = 5x 3 + x 2 - 9x + 4
y = 3x(x - 1)(x - 1)
Factor the quadratic.
28. f (x) = x 4 - 4x - 1
The zeros are 0, and 1 with multiplicity 2.
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Topic 8
Review
29. f (x) = x 3 - 3x 2 - 3x - 4
8-4 Solving Polynomial Equations
Quick Review
Exercises
One way to solve a polynomial equation is by factoring.
First write the equation in the form P (x) = 0, where P (x)
is the polynomial. Then factor the polynomial. Last, use
the Zero-Product Property to find the solutions, or roots.
The solutions may be real or imaginary. Real solutions and
approximations of irrational solutions can also be found by
using a graphing calculator.
Find the real or imaginary solutions of each equation
by factoring.
30. x 2 - 11x = -24
31. 4x 2 = -4x - 1
32. 3x 3 + 3x 2 = 27x
33. 2x 2 + 3 = 4x
Find the real roots of each equation by graphing.
34. x 4 + 3x 2 - 2x + 5 = 0
Example
Solve
x3
x3
+
+
35. x 2 + 3 = x 3 - 5
4x 2
4x 2
= 12x by factoring.
- 12x = 0
Subtract 12x from each side.
x(x - 2)(x + 6) = 0
Factor the left side.
x = 0, x - 2 = 0, x + 6 = 0
Zero-Product Property
x = 0, x = 2, x = - 6
Solve each equation.
36. The height and width of a rectangular prism are each
2 inches shorter than the length of the prism. The
volume of the prism is 40 cubic inches. Approximate the
dimensions of the prism to the nearest hundredth.
The solutions are 0, 2, and -6.
8-5 Dividing Polynomials
Quick Review
Exercises
You can divide a polynomial by one of its factors to find
another factor. When you divide by a linear factor, you can
simplify this division by writing only the coefficients of each
term. This is called synthetic division. The Remainder
Theorem says that P (a) is the remainder when you divide
P (x) by x - a.
Divide using long division. Check your answers.
37. (x 3 + 7x 2 + 15x + 9) , (x + 1)
38. (2x 3 - 7x 2 - 7x + 13) , (x - 4)
Determine whether each binomial is a factor of
x 3 + x 2 − 10x + 8.
Example
39. x - 2
Let P (x) = 3x 2 - 13x + 15. What is P (3)?
Divide using synthetic division.
According to the Remainder Theorem, P (3) is the
remainder when you divide P (x) by x - 3.
41. (x 3 + 5x 2 - x - 5) , (x + 5)
3
3
⫺13
15
9 ⫺12
3
⫺4
Put the opposite of the
constant in the divisor
at the top left.
3
The quotient is 3x - 4 with remainder 3, so P (3) = 3.
40. x - 4
42. (2x 3 + 14x 2 - 58x) , (x + 10)
43. (5x 3 + 8x 2 - 60) , (x - 2)
Use the Remainder Theorem to determine the value
of P (a).
44. P (x) = 2x 3 + 5x 2 + 7x - 4, a = -2
45. P (x) = x 3 - 4x 2 + 2x + 3, a = 1
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8-6 Theorems About Roots of Polynomial Equations
Quick Review
Exercises
The Rational Root Theorem gives a way to determine
the possible roots of a polynomial equation P (x) = 0. If the
coefficients of P(x) are all integers, then every root of the
p
equation can be written in the form q , where p is a factor of
the constant term and q is a factor of the leading coefficient.
List the possible rational roots of P (x) given by the
Rational Root Theorem.
The Conjugate Root Theorem states that if P(x) is a
polynomial with rational coefficients, then irrational
roots that have the form a + 1b and imaginary roots of
P (x) = 0 come in conjugate pairs. Therefore, if a + 1b is
an irrational root, where a and b are rational, then a - 1b
is also a root. Likewise, if a + bi is a root, where a and b are
real and i is the imaginary unit, then a - bi is also a root.
48. P (x) = 4x 4 - 2x 3 + x 2 - 12
Descartes’ Rule of Signs gives a way to determine the
possible number of positive and negative real roots by
analyzing the signs of the coefficients. The number of
positive real roots is equal to the number of sign changes
in consecutive coefficients of P(x), or is less than that by an
even number. The number of negative real roots is equal to
the number of sign changes in consecutive coefficients of
P( -x), or is less than that by an even number.
46. P(x) = x 3 + 4x 2 - 10x + 6
47. P (x) = 3x 3 - x 2 - 7x + 2
49. P (x) = 3x 4 - 4x 3 - x 2 - 7
Find any rational roots of P (x).
50. P(x) = x 3 + 2x 2 + 4x + 21
51. P(x) = x 3 + 5x 2 + x + 5
52. P(x) = 2x 3 + 7x 2 - 5x - 4
53. P(x) = 3x 4 + 2x 3 - 9x 2 + 4
A polynomial P (x) has rational coefficients. Name
additional roots of P (x) given the following roots.
54. 1 - i and 5
55. 5 + 13 and - 12
Example
56. -3i and 7i
Find the rational roots of P (x) = 0 if
P (x) = 2x 3 - 4x 2 - 10x + 12.
57. -2 + 211 and -4 - 6i
List the possible roots: { 12, {1, { 32, {2, {3, {4, {6, {12.
Use synthetic division to test roots.
Write a polynomial function with the given roots.
3
2
2
⫺4
⫺10
12
6
6
⫺12
2
⫺4
0
So x - 3 and (2x 2 + 2x - 4) are factors of P(x).
P (x) = (x -
3)(2x 2
+ 2x - 4)
Factor the quadratic.
P (x) = 2(x - 3)(x + 2)(x - 1)
Solve 2(x - 3)(x + 2)(x - 1) = 0.
x = 3, x = - 2, or x = 1
The rational roots are 3, -2, and 1.
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Topic 8 5HYLHZ
58. 7 and 10
59. -3 and 5i
60. 6 - i
61. 3 + i, 2, and -4
Determine the possible number of positive real zeros and
negative real zeros for each polynomial function given by
Descartes’ Rule of Signs.
62. P (x) = 5x 3 + 7x 2 - 2x - 1
63. P (x) = -3x 3 + 11x 2 + 12x - 8
64. P (x) = 6x 4 - x 3 + 5x 2 - x + 9
65. P (x) = -x 4 - 3x 3 + 8x 2 + 2x - 14
8-7 The Fundamental Theorem of Algebra
Quick Review
Exercises
The Fundamental Theorem of Algebra states that if P (x)
is a polynomial of degree n, where n Ú 1, then P (x) = 0 has
exactly n roots. This includes multiple and complex roots.
Find the number of roots for each equation.
Example
68. -x 5 - 6 = 0
Use the Fundamental Theorem of Algebra to determine the
number of roots for x 4 + 2x 2 - 3 = 0.
69. 5x 4 - 7x 6 + 2x 3 + 8x 2 + 4x - 11 = 0
Because the polynomial is of degree 4, it has 4 roots.
66. x 3 - 2x + 5 = 0
67. 2 - x 4 + x 2 = 0
Find all the zeros for each function.
70. P (x) = x 3 + 5x 2 - 4x - 2
71. P (x) = x 4 - 4x 3 - x 2 + 20x - 20
72. P (x) = 2x 3 - 3x 2 + 3x - 2
73. P (x) = x 4 - 4x 3 - 16x 2 + 21x + 18
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