230 CHAPTER 4 Polynomial and Rational Functions NOTE: Remember, the real numbers are a subset of the set of complex numbers. So, a complex zero is not necessarily a real zero, but every real zero is a complex zero. For example, − =− + i 3 3 0 is a complex zero that is real, and − + i 3 5 is a complex zero that is not real. j Fundamental Theorem of Algebra Every complex polynomial function f of degree ≥ n 1 has at least one complex zero. Proof Let ( ) = + + + + − − f x a x a x a x a n n n n 1 1 1 0 . By the Fundamental Theorem of Algebra, f has at least one zero, say r .1 Then, by the Factor Theorem, −x r1 is a factor, and ( ) ( ) ( ) = − f x x r q x 1 1 where ( ) q x 1 is a complex polynomial of degree −n 1 whose leading coefficient is a .n Repeating this argument n times, we arrive at ( ) ( )( ) ( ) ( ) = − − ⋅ ⋅ − f x x r x r x r q x n n 1 2 where ( ) q x n is a complex polynomial of degree − = n n 0 whose leading coefficient is a .n That is, ( ) = = q x a x a , n n n 0 and so ( ) ( )( ) ( ) = − − ⋅ ⋅ − f x a x r x r x r n n 1 2 We conclude that every complex polynomial function f of degree ≥ n 1 has exactly n (not necessarily distinct) zeros. ■ In Section 4.3, we found the real zeros of polynomial functions of degree 3 or higher. In this section we will find the complex zeros of polynomial functions of degree 3 or higher. DEFINITION Complex Zeros A variable in the complex number system is referred to as a complex variable . A complex polynomial function f of degree n is a function of the form ( ) = + + + + − − f x a x a x a x a n n n n 1 1 1 0 (1) where − a a a a , , . . . , , n n 1 1 0 are complex numbers, ≠ a 0, n n is a nonnegative integer, and x is a complex variable.As before, an is called the leading coefficient of f . A complex number r is called a complex zero of f if ( ) = f r 0. In most of our work, the coefficients in (1) are real numbers. We have learned that some quadratic equations have no real solutions, but that in the complex number system every quadratic equation has at least one solution, either real or complex. The next result, proved by Karl Friedrich Gauss (1777–1855) when he was 22 years old,* gives an extension to complex polynomials. In fact, this result is so important and useful that it has become known as the Fundamental Theorem of Algebra . We shall not prove this result, as the proof is beyond the scope of this text. However, using the Fundamental Theorem of Algebra and the Factor Theorem, we can prove the following result: THEOREM Every complex polynomial function f of degree ≥ n 1 can be factored into n linear factors (not necessarily distinct) of the form ( ) ( )( ) ( ) = − − ⋅ ⋅ − f x a x r x r x r n n 1 2 (2) where … a r r r , , , , n n 1 2 are complex numbers.That is, every complex polynomial function of degree ≥ n 1 has exactly n complex zeros, some of which may repeat. *In all, Gauss gave four different proofs of this theorem, the first one in 1799 being the subject of his doctoral dissertation. Need to Review? Complex numbers and the complex solutions of a quadratic equation are discussed in Section A.7, pp. A58 – A6 5 .
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