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Unformatted text preview: Section 3.6 Complex zeros of Polynomials The Fundamental Theorem of Algebra. If f ( x ) = a n x n + a n- 1 x n- 1 + + a 1 x + a , then f ( x ) = 0 for some number, x. (The number x guaranteed by the Fundamental Theorem may be a complex (non real) number.) Note. The Fundamental Theorem implies that an n-th degree polynomial can always be factored into n linear factor, if complex numbers are allowed in the linear factors. Factoring. If r 1 , r 2 , , r n are the zeros of a polynomial, f , then f ( x ) = a n x n + a n- 1 x n- 1 + + a 1 x + a = a n ( x- r 1 )( x- r 2 ) ( x- r n ) Note. In the above, the coefficients, a j , and the zeros, r j , can be complex (non-real) numbers. Recall that when a polynomial with real coefficients has a complex zero, then the complex conjugate of this zero is also a zero of the polynomial. Example. Form a polynomial f ( x ) with a real coefficients having degree 5 and zeros at 2- i , 4 i , and 3....
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