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SECTION 3.6 COMPLEX ZEROS; COMPLEX ZEROS; FUNDAMENTAL THEOREM OF ALGEBRA FUNDAMENTAL THEOREM OF ALGEBRA

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COMPLEX POLYNOMIAL FUNCTION A complex polynomial function f of degree n is a complex function of the form f(x) = a n x n + a n-1 x n-1 +... + a 1 x + a 0 where a n, a n-1,..., a 1, a 0 are complex numbers, a n 0, n is a nonnegative integer, and x is a complex variable.

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COMPLEX ZERO A complex number r is called a complex zero of a complex function f if f(r) = 0.

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COMPLEX ZEROS We have learned that some quadratic equations have no real solutions but that in the complex number system every quadratic equation has a solution, either real or complex.

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FUNDAMENTAL THEOREM OF ALGEBRA Every complex polynomial function f(x) of degree n 1 has at least one complex zero.

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THEOREM Every complex polynomial function f(x) of degree n 1 can be factored into n linear factors (not necessarily distinct) of the form f(x) = a n (x - r 1 )(x - r 2 ) (x - r n ) where a n, r 1, r 2,..., r n are complex numbers.

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CONJUGATE PAIRS THEOREM Let f(x) be a complex polynomial whose coefficients are real numbers. If r = a + bi is a zero of f, then the complex conjugate r = a - bi is also a zero of f.

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CONJUGATE PAIRS THEOREM In other words, for complex polynomials whose coefficients are real numbers, the zeros occur in conjugate pairs.

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CORORLLARY A complex polynomial f of odd degree with real coefficients has at least one real zero.

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EXAMPLE A polynomial f of degree 5 whose coefficients are real numbers has the zeros 1, 5i, and 1 + i. Find the remaining two zeros. - 5i 1 - i

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EXAMPLE Find a polynomial f of degree 4 whose coefficients are real numbers and has the zeros 1, 1, and - 4 + i. f(x) = a(x - 1)(x - 1)[x - (- 4 + i)][x - (- 4 - i)] First, let a = 1; Graph the resulting polynomial. Then look at other a’s.

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EXAMPLE It is known that 2 + i is a zero of f(x) = x 4 - 8x 3 + 64x - 105 Find the remaining zeros. - 3, 7, 2 + i and 2 - i

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EXAMPLE Find the complex zeros of the polynomial function f(x) = 3x 4 + 5x 3 + 25x 2 + 45x - 18

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CONCLUSION OF SECTION 3.6 CONCLUSION OF SECTION 3.6

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