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ss_3_6

# ss_3_6 - Section 3.6 Complex zeros of Polynomials The...

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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 0 , 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 0 = 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. Solution. Since the coefficients are real, the complex conjugates of 2 - i and 4 i must also be zeros.

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ss_3_6 - Section 3.6 Complex zeros of Polynomials The...

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