Stewart Precalc 6e Section 3.6: Complex Zeros and the Fundamental Theorem of Algebra

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Stewart Precalc 6e Section 3.6: Complex Zeros and the Fundamental Theorem of Algebra 0/83
1 Concepts · Level 1
The polynomial \(P(x) = 3(x - 5)^3 (x - 3)(x + 2)\) has degree _____. It has zeros \(5\), \(3\), and _____. The zero \(5\) has multiplicity _____, and the zero \(3\) has multiplicity _____.
2 Concepts · Level 1
(a) If \(a\) is a zero of the polynomial \(P\), then _____ must be a factor of \(P(x)\).
(b) If \(a\) is a zero of multiplicity \(m\) of the polynomial \(P\), then _____ must be a factor of \(P(x)\) when we factor \(P\) completely.

Enter your answer directly below each part above.

3 Concepts · Level 1
A polynomial of degree \(n \geq 1\) has exactly _____ zeros if a zero of multiplicity \(m\) is counted \(m\) times.
4 Concepts · Level 1
If the polynomial function \(P\) has real coefficients and if \(a + b i\) is a zero of \(P\), then _____ is also a zero of \(P\).
5 Skills - Find Zeros and Factor · Level 2
\( P(x) = x^4 + 4 x^2 \)
6 Skills - Find Zeros and Factor · Level 2
\( P(x) = x^5 + 9 x^3 \)
7 Skills - Find Zeros and Factor · Level 2
\( P(x) = x^3 - 2 x^2 + 2 x \)
8 Skills - Find Zeros and Factor · Level 2
\( P(x) = x^3 + x^2 + x \)
9 Skills - Find Zeros and Factor · Level 2
\( P(x) = x^4 + 2 x^2 + 1 \)
10 Skills - Find Zeros and Factor · Level 2
\( P(x) = x^4 - x^2 - 2 \)
11 Skills - Find Zeros and Factor · Level 2
\( P(x) = x^4 - 16 \)
12 Skills - Find Zeros and Factor · Level 2
\( P(x) = x^4 + 6 x^2 + 9 \)
13 Skills - Find Zeros and Factor · Level 3
\( P(x) = x^3 + 8 \)
14 Skills - Find Zeros and Factor · Level 3
\( P(x) = x^3 - 8 \)
15 Skills - Find Zeros and Factor · Level 3
\( P(x) = x^6 - 1 \)
16 Skills - Find Zeros and Factor · Level 3
\( P(x) = x^6 - 7 x^3 - 8 \)
17 Skills - Factor and State Multiplicities · Level 1
\( P(x) = x^2 + 25 \)
18 Skills - Factor and State Multiplicities · Level 1
\( P(x) = 4 x^2 + 9 \)
19 Skills - Factor and State Multiplicities · Level 2
\( Q(x) = x^2 + 2 x + 2 \)
20 Skills - Factor and State Multiplicities · Level 2
\( Q(x) = x^2 - 8 x + 17 \)
21 Skills - Factor and State Multiplicities · Level 2
\( P(x) = x^3 + 4 x \)
22 Skills - Factor and State Multiplicities · Level 2
\( P(x) = x^3 - x^2 + x \)
23 Skills - Factor and State Multiplicities · Level 2
\( Q(x) = x^4 - 1 \)
24 Skills - Factor and State Multiplicities · Level 2
\( Q(x) = x^4 - 625 \)
25 Skills - Factor and State Multiplicities · Level 2
\( P(x) = 16 x^4 - 81 \)
26 Skills - Factor and State Multiplicities · Level 3
\( P(x) = x^3 - 64 \)
27 Skills - Factor and State Multiplicities · Level 3
\( P(x) = x^3 + x^2 + 9 x + 9 \)
28 Skills - Factor and State Multiplicities · Level 4
\( P(x) = x^6 - 729 \)
29 Skills - Factor and State Multiplicities · Level 2
\( Q(x) = x^4 + 2 x^2 + 1 \)
30 Skills - Factor and State Multiplicities · Level 2
\( Q(x) = x^4 + 10 x^2 + 25 \)
31 Skills - Factor and State Multiplicities · Level 2
\( P(x) = x^4 + 3 x^2 - 4 \)
32 Skills - Factor and State Multiplicities · Level 2
\( P(x) = x^5 + 7 x^3 \)
33 Skills - Factor and State Multiplicities · Level 3
\( P(x) = x^5 + 6 x^3 + 9 x \)
34 Skills - Factor and State Multiplicities · Level 4
\( P(x) = x^6 + 16 x^3 + 64 \)
35 Skills - Polynomial with Given Zeros · Level 2
\(P\) has degree 2 and zeros \(1 + i\) and \(1 - i\).
36 Skills - Polynomial with Given Zeros · Level 2
\(P\) has degree 2 and zeros \(1 + i \sqrt{2}\) and \(1 - i \sqrt{2}\).
37 Skills - Polynomial with Given Zeros · Level 2
\(Q\) has degree 3 and zeros \(3\), \(2 i\), and \(-2 i\).
38 Skills - Polynomial with Given Zeros · Level 2
\(Q\) has degree 3 and zeros \(0\) and \(i\).
39 Skills - Polynomial with Given Zeros · Level 2
\(P\) has degree 3 and zeros \(2\) and \(i\).
40 Skills - Polynomial with Given Zeros · Level 2
\(Q\) has degree 3 and zeros \(-3\) and \(1 + i\).
41 Skills - Polynomial with Given Zeros · Level 3
\(R\) has degree 4 and zeros \(1 - 2 i\) and \(1\), with \(1\) a zero of multiplicity 2.
42 Skills - Polynomial with Given Zeros · Level 3
\(S\) has degree 4 and zeros \(2 i\) and \(3 i\).
43 Skills - Polynomial with Given Zeros · Level 4
\(T\) has degree 4, zeros \(i\) and \(1 + i\), and constant term \(12\).
44 Skills - Polynomial with Given Zeros · Level 4
\(U\) has degree 5, zeros \(\dfrac{1}{2}\), \(-1\), and \(-i\), and leading coefficient \(4\); the zero \(-1\) has multiplicity 2.
45 Skills - Find All Zeros · Level 2
\( P(x) = x^3 + 2 x^2 + 4 x + 8 \)
46 Skills - Find All Zeros · Level 3
\( P(x) = x^3 - 7 x^2 + 17 x - 15 \)
47 Skills - Find All Zeros · Level 3
\( P(x) = x^3 - 2 x^2 + 2 x - 1 \)
48 Skills - Find All Zeros · Level 3
\( P(x) = x^3 + 7 x^2 + 18 x + 18 \)
49 Skills - Find All Zeros · Level 3
\( P(x) = x^3 - 3 x^2 + 3 x - 2 \)
50 Skills - Find All Zeros · Level 3
\( P(x) = x^3 - x - 6 \)
51 Skills - Find All Zeros · Level 3
\( P(x) = 2 x^3 + 7 x^2 + 12 x + 9 \)
52 Skills - Find All Zeros · Level 3
\( P(x) = 2 x^3 - 8 x^2 + 9 x - 9 \)
53 Skills - Find All Zeros · Level 4
\( P(x) = x^4 + x^3 + 7 x^2 + 9 x - 18 \)
54 Skills - Find All Zeros · Level 4
\( P(x) = x^4 - 2 x^3 - 2 x^2 - 2 x - 3 \)
55 Skills - Find All Zeros · Level 4
\( P(x) = x^5 - x^4 + 7 x^3 - 7 x^2 + 12 x - 12 \)
56 Skills - Find All Zeros · Level 4
\(P(x) = x^5 + x^3 + 8 x^2 + 8\) [Hint: Factor by grouping.]
57 Skills - Find All Zeros · Level 4
\( P(x) = x^4 - 6 x^3 + 13 x^2 - 24 x + 36 \)
58 Skills - Find All Zeros · Level 4
\( P(x) = x^4 - x^2 + 2 x + 2 \)
59 Skills - Find All Zeros · Level 4
\( P(x) = 4 x^4 + 4 x^3 + 5 x^2 + 4 x + 1 \)
60 Skills - Find All Zeros · Level 4
\( P(x) = 4 x^4 + 2 x^3 - 2 x^2 - 3 x - 1 \)
61 Skills - Find All Zeros · Level 4
\( P(x) = x^5 - 3 x^4 + 12 x^3 - 28 x^2 + 27 x - 9 \)
62 Skills - Find All Zeros · Level 4
\( P(x) = x^5 - 2 x^4 + 2 x^3 - 4 x^2 + x - 2 \)
63 Skills - Linear and Quadratic Factors · Level 3
(a) Factor \(P\) into linear and irreducible quadratic factors with real coefficients. (b) Factor \(P\) completely into linear factors with complex coefficients. \(P(x) = x^3 - 5 x^2 + 4 x - 20\)
64 Skills - Linear and Quadratic Factors · Level 3
(a) Factor \(P\) into linear and irreducible quadratic factors with real coefficients. (b) Factor \(P\) completely into linear factors with complex coefficients. \(P(x) = x^3 - 2 x - 4\)
65 Skills - Linear and Quadratic Factors · Level 2
(a) Factor \(P\) into linear and irreducible quadratic factors with real coefficients. (b) Factor \(P\) completely into linear factors with complex coefficients. \(P(x) = x^4 + 8 x^2 - 9\)
66 Skills - Linear and Quadratic Factors · Level 2
(a) Factor \(P\) into linear and irreducible quadratic factors with real coefficients. (b) Factor \(P\) completely into linear factors with complex coefficients. \(P(x) = x^4 + 8 x^2 + 16\)
67 Skills - Linear and Quadratic Factors · Level 4
(a) Factor \(P\) into linear and irreducible quadratic factors with real coefficients. (b) Factor \(P\) completely into linear factors with complex coefficients. \(P(x) = x^6 - 64\)
68 Skills - Linear and Quadratic Factors · Level 2
(a) Factor \(P\) into linear and irreducible quadratic factors with real coefficients. (b) Factor \(P\) completely into linear factors with complex coefficients. \(P(x) = x^5 - 16 x\)
69 Applications - Real and Imaginary Solutions · Level 3
By the Zeros Theorem, every \(n\)th-degree polynomial equation has exactly \(n\) solutions (including possibly some that are repeated). Some of these may be real, and some may be imaginary. Use a graphing device to determine how many real and imaginary solutions each equation has.
(a) \(x^4 - 2 x^3 - 11 x^2 + 12 x = 0\)
(b) \(x^4 - 2 x^3 - 11 x^2 + 12 x - 5 = 0\)
(c) \(x^4 - 2 x^3 - 11 x^2 + 12 x + 40 = 0\)

Enter your answer directly below each part above.

70 Applications - Imaginary Coefficients · Level 3
Find all solutions of the equation.
(a) \(2 x + 4 i = 1\)
(b) \(x^2 - i x = 0\)
(c) \(x^2 + 2 i x - 1 = 0\)
(d) \(i x^2 - 2 x + i = 0\)

Enter your answer directly below each part above.

71 Applications - Imaginary Coefficients · Level 4
(a) Show that \(2 i\) and \(1 - i\) are both solutions of the equation \(x^2 - (1 + i) x + (2 + 2 i) = 0\) but that their complex conjugates \(-2 i\) and \(1 + i\) are not.
(b) Explain why the result of part (a) does not violate the Conjugate Zeros Theorem.

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72 Applications - Imaginary Coefficients · Level 4
(a) Find the polynomial with real coefficients of the smallest possible degree for which \(i\) and \(1 + i\) are zeros and in which the coefficient of the highest power is \(1\).
(b) Find the polynomial with complex coefficients of the smallest possible degree for which \(i\) and \(1 + i\) are zeros and in which the coefficient of the highest power is \(1\).

Enter your answer directly below each part above.

73 Discovery - Polynomials of Odd Degree · Level 3
Polynomials of Odd Degree. The Conjugate Zeros Theorem says that the complex zeros of a polynomial with real coefficients occur in complex conjugate pairs. Explain how this fact proves that a polynomial with real coefficients and odd degree has at least one real zero.
74 Discovery - Roots of Unity · Level 3
Roots of Unity. There are two square roots of \(1\), namely, \(1\) and \(-1\). These are the solutions of \(x^2 = 1\). The fourth roots of \(1\) are the solutions of the equation \(x^4 = 1\) or \(x^4 - 1 = 0\). How many fourth roots of \(1\) are there? Find them. The cube roots of \(1\) are the solutions of the equation \(x^3 = 1\) or \(x^3 - 1 = 0\). How many cube roots of \(1\) are there? Find them. How would you find the sixth roots of \(1\)? How many are there? Make a conjecture about the number of \(n\)th roots of \(1\).
75 Example - Factoring a Polynomial Completely · Level 3
Let \(P(x) = x^3 - 3 x^2 + x - 3\). (a) Find all the zeros of \(P\). (b) Find the complete factorization of \(P\).
76 Example - Factoring a Polynomial Completely · Level 3
Let \(P(x) = x^3 - 2 x + 4\). (a) Find all the zeros of \(P\). (b) Find the complete factorization of \(P\).
77 Example - Factoring a Polynomial with Complex Zeros · Level 3
Find the complete factorization and all five zeros of the polynomial \(P(x) = 3 x^5 + 24 x^3 + 48 x\).
78 Example - Finding Polynomials with Specified Zeros · Level 3
(a) Find a polynomial \(P(x)\) of degree 4, with zeros \(i\), \(-i\), \(2\), and \(-2\), and with \(P(3) = 25\).
(b) Find a polynomial \(Q(x)\) of degree 4, with zeros \(-2\) and \(0\), where \(-2\) is a zero of multiplicity 3.

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79 Example - Finding All the Zeros of a Polynomial · Level 4
Find all four zeros of \(P(x) = 3 x^4 - 2 x^3 - x^2 - 12 x - 4\).
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80 Example - Polynomial with a Specified Complex Zero · Level 3
Find a polynomial \(P(x)\) of degree 3 that has integer coefficients and zeros \(\dfrac{1}{2}\) and \(3 - i\).
81 Example - Factoring into Linear and Quadratic Factors · Level 3
Let \(P(x) = x^4 + 2 x^2 - 8\).
(a) Factor \(P\) into linear and irreducible quadratic factors with real coefficients.
(b) Factor \(P\) completely into linear factors with complex coefficients.

Enter your answer directly below each part above.

82 Example - Rational Function with a Slant Asymptote · Level 3
Graph the rational function \(r(x) = \dfrac{x^2 - 4x - 5}{x - 3}\).
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83 Example - End Behavior of a Rational Function · Level 4
Graph the rational function \(r(x) = \dfrac{x^3 - 2x^2 + 3}{x - 2}\) and describe its end behavior.

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