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

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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 can’t 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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