MAT/116
3.5

# B use the intermediate value theorem to find an

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b. Use the Intermediate Value Theorem to find an approximation for this real zero to the nearest tenth The sign change indicates that f has a real zero between 2 and 2.1. x f ( x ) = x 3 - 2 x - 5 2 f (2) = 2 3 - 2(2) - 5 = - 1 2.1 f (2.1) = (2.1) 3 - 2(2.1) - 5 = 0.061 Sign change Sign change 3.5: More on Zeros of Polynomial Functions more

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f (2.00) = - 1 f (2.04) = - 0.590336 f (2.08) = - 0.161088 f (2.01) = - 0.899399 f (2.05) = - 0.484875 f (2.09) = - 0.050671 f (2.02) = - 0.797592 f (2.06) = - 0.378184 f (2.1) = 0.061 f (2.03) = - 0.694573 f (2.07) = - 0.270257 EXAMPLE : Approximating a Real Zero b. We now follow a similar procedure to locate the real zero between successive hundredths. We divide the interval [2, 2.1] into ten equal sub- intervals. Then we evaluate f at each endpoint and look for a sign change. Solution a. Show that the polynomial function f ( x ) = x 3 - 2 x - 5 has a real zero between 2 and 3. b. Use the Intermediate Value Theorem to find an approximation for this real zero to the nearest tenth The sign change indicates that f has a real zero between 2.09 and 2.1. Correct to the nearest tenth, the zero is 2.1. Sign change 3.5: More on Zeros of Polynomial Functions
3.5: More on Zeros of Polynomial Functions We have seen that if a polynomial equation is of degree n, then counting multiple roots separately, the equation has n roots. This result is called the Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra The Fundamental Theorem of Algebra If f ( x ) is a polynomial of degree n, where n 1, then the equation f ( x ) = 0 has at least one complex root. The Fundamental Theorem of Algebra The Fundamental Theorem of Algebra

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The Linear Factorization Theorem The Linear Factorization Theorem The Linear Factorization Theorem The Linear Factorization Theorem If f ( x ) = a n x n + a n - 1 x n - 1 + + a 1 x + a 0 b, where n 1 and a n 0 , then f ( x ) = a n ( x - c 1 ) ( x - c 2 ) ( x - c n ) where c 1 , c 2 ,…, c n are complex numbers (possibly real and not necessarily distinct). In words: An n th-degree polynomial can be expressed as the product of n linear factors.
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