Mth141S37n - Sec. 3.7 Topics in the Theory of Polynomial...

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Unformatted text preview: Sec. 3.7 Topics in the Theory of Polynomial Functions (II) In section 3.7, we continue with information that will help us find the zeros of a polynomial function. Remember the real zeros are the x-intercepts of the graph. The Conjugate Zeros Theorem says : if you have a polynomial function with real coefficients and a + bi is a zero of the function, then its conjugate a bi must also be a zero of the function. What this really says is that imaginary zeros always come in pairs!!!!!!! You cant have just one imaginary zero !!! (Or 3, or 5, or ) You must always have an even number of imaginary zeros. Ex. If some zeros of a polynomial function with real coefficients are -1, 2, and 3 + 2i, can you find any other zeros. Yes! Since 3 + 2i is a zero, its conjugate 3 2i must also be a zero. Ex. Find a polynomial function with real coefficients that has zeros of -1, 2, and 3 + 2i. First, we know 3 2i is also a zero. Second, if we know the zeros, then we should know the factors needed. So zero -1 (x + 1) is a factor zero 2 (x 2) is a factor zero 3 + 2i (x 3 2i) is a factor zero 3 2i (x 3 + 2i) is a factor So a polynomial f(x) = (x + 1)(x 2)(x 3 2i)(x 3 + 2i) should have those zeros. = 2 2 ( 2)( 6 13) x x x x--- + = 4 3 2 7 17 26 x x x x- +-- This, of course, is only one such polynomial that has these zeros. We could find infinitely more of them if we multiply through by a number (-1, 2, 3, 4,).- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - The Fundamental Theorem of Algebra says that every polynomial function of degree 1 has at least one complex zero....
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Mth141S37n - Sec. 3.7 Topics in the Theory of Polynomial...

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