SECTION 4.4 Complex Zeros; Fundamental Theorem of Algebra 231 Conjugate Pairs Theorem Suppose f is a polynomial function whose coefficients are real numbers. If = + r a bi is a zero of f , the complex conjugate = − r a bi is also a zero of f . Corollary A polynomial function f of odd degree with real coefficients has at least one real zero. Proof Because complex zeros occur as conjugate pairs in a polynomial function with real coefficients, there will always be an even number of zeros that are not real numbers. Consequently, since f is of odd degree, one of its zeros must be a real number. ■ 1 Use the Conjugate Pairs Theorem The Fundamental Theorem of Algebra can be used to obtain valuable information about the complex zeros of polynomial functions whose coefficients are real numbers. In other words, for polynomial functions whose coefficients are real numbers, the complex zeros occur in conjugate pairs. This result should not be all that surprising since the complex zeros of a quadratic function occurred in conjugate pairs. Proof Let ( ) = + + + + − − f x a x a x a x a n n n n 1 1 1 0 where − a a a a , , . . . , , n n 1 1 0 are real numbers and ≠ a 0. n If = + r a bi is a zero of f , then ( ) ( ) = + = f r f a bi 0, so + + + + = − − a r a r a r a 0 n n n n 1 1 1 0 Take the conjugate of both sides to get ( ) ( ) ( ) ( ) + + + + = + + + + = + + + + = + + + + = − − − − − − − − a r a r a r a a r a r a r a a r a r a r a a r a r a r a 0 0 0 0 n n n n n n n n n n n n n n n n 1 1 1 0 1 1 1 0 1 1 1 0 1 1 1 0 This last equation states that ( ) = f r 0; that is, = − r a bi is a zero of f . ■ The conjugate of a sum equals the sum of the conjugates (see Section A.7). The conjugate of a product equals the product of the conjugates. The conjugate of a real number equals the real number. The importance of this result is that once we know a complex number, say + i 3 4 , is a zero of a polynomial function with real coefficients, then we know that its conjugate − i 3 4 is also a zero. This result has an important corollary. For example, the polynomial function ( ) = − + − f x x x x 3 4 5 5 4 3 has at least one zero that is a real number, since f is of degree 5 (odd) and has real coefficients.
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