6-6 The Fundamental Theorem of Algebra

6-6 The Fundamental Theorem of Algebra Objectives The Fundamental Theorem of Algebra Vocabulary Fundamental Theorem of Algebra If P(x) is a polynomial of degree n 1 with complex coefficients, then P(x) = 0 has at least one complex root.

Corollary Including complex roots and multiple roots, an nth degree polynomial equation has exactly n roots; the related polynomial function has exactly n zeros. For the equation x4 3x3 + 4x + 1 = 0, find the number of complex roots, the possible number of real roots, and the possible rational roots. By the corollary to the Fundamental Theorem of Algebra, x4 3x3 + 4x + 1 = 0 has four complex roots. By the Imaginary Root Theorem, the equation has either no imaginary roots,

two imaginary roots (one conjugate pair), or four imaginary roots (two conjugate pairs). So the equation has either zero real roots, two real roots, or four real roots. By the Rational Root Theorem, the possible rational roots of the equation are 1. Find the number of complex zeros of (x) = x5 + 3x4 x 3. Find all the zeros. By the corollary to the Fundamental Theorem of Algebra, there are five complex zeros. You can use synthetic division to find a rational zero. Step 1: Find a rational root from the possible roots of 1 and 3.

Use synthetic division to test each possible until you get a remainder of zero. 3 1 1 x4 3 0 3 0 0 0 0

0 0 1 3 0 3 1 0 1 So 3 is one of the roots. (continued)

Step 2: Factor the expression x4 1. x4 1 = (x2 1)(x2 + 1) = (x 1)(x + 1)(x2 + 1) So 1 and 1 are also roots. Factor x2 1. Step 3: Solve x2 + 1 = 0. x2 + 1 = 0 x2 = 1

x =i So i and i are also roots. The polynomial function (x) = x5 + 3x4 x 3 has three real zeros of x = 3, x = 1, and x = 1, and two complex zeros of x = i, and x = i. Homework Pg 343 # 1, 2, 9, 10, 17, 18

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