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Sullivan Algebra and Trigonometry: Section 5.6 Complex Zeros; Fundamental Theorem of Algebra Objectives Utilize the Conjugate Pairs Theorem to Find the Complex Zeros of a Polynomial Find a Polynomial Function with Specified Zeros Find the Complex Zeros of a Polynomial

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A variable in the complex number system is referred to as a complex variable. A complex polynomial function f degree n is a complex function of the form A complex number r is called a (complex) zero of a complex function f if f (r) = 0. Note that real numbers are also complex numbers in the form a + bi where b = 0. So, this definition of a complex polynomial function is a generalization of what was previously introduced.

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Fundamental Theorem of Algebra Every complex polynomial function f (x) of degree n > 1 has at least one complex zero. Theorem Every complex polynomial function f (x) of degree n > 1 can be factored into n linear factors (not necessarily distinct) of the form

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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 is also a zero of f. Corollary A complex polynomial f of odd degree with real coefficients has at least one real zero.

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Find a polynomial f of degree 4 whose coefficients are real and has zeros 0, -2 and 1 - 3i. Graph f to verify the solution.

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Find the complex zeros of the polynomial There are 4 complex zeros. Using Descartes’ Rule of Sign, there are two, or no, positive real zeros. Using Descartes’ Rule of Sign, there are two, or no, negative real zeros.

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Now, list all possible rational zeros p/q by factoring the first and last coefficients of the function. Now, begin testing each potential zero using synthetic division. If a potential zero k is in fact a zero, then x - k divides into f (remainder will be zero) and is a factor of f.

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Test k = -2 Thus, -2 is a zero of f and x + 2 is a factor of f. Test k = -1/2 Thus, -1/2 is a zero of f and x + 1/2 is a factor of f.

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Now, find the complex zeros of the quadratic factor.

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