Just as an n th degree polynomial equation has n

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Just as an n th-degree polynomial equation has n roots, an n th-degree polynomial has n linear factors. This is formally stated as the Linear Factorization Theorem. 3.5: More on Zeros of Polynomial Functions
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3.5: More on Zeros of Polynomial Functions EXAMPLE: Finding a Polynomial Function with Given Zeros Find a fourth-degree polynomial function f ( x ) with real coefficients that has -2, 2, and i as zeros and such that f (3) = - 150. Solution Because i is a zero and the polynomial has real coefficients, the conjugate must also be a zero. We can now use the Linear Factorization Theorem. = a n ( x + 2)( x - 2)( x - i )( x + i ) Use the given zeros: c 1 = - 2, c 2 = 2, c 3 = i , and, from above, c 4 = - i . f ( x ) = a n ( x - c 1 )( x - c 2 )( x - c 3 )( x - c 4 ) This is the linear factorization for a fourth- degree polynomial. = a n ( x 2 - 4)( x 2 + i ) Multiply f ( x ) = a n ( x 4 - 3 x 2 - 4) Complete the multiplication more more
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3.5: More on Zeros of Polynomial Functions EXAMPLE: Finding a Polynomial Function with Given Zeros Find a fourth-degree polynomial function f ( x ) with real coefficients that has -2, 2, and i as zeros and such that f (3) = - 150. Substituting - 3 for a n in the formula for f ( x ) , we obtain f ( x ) = - 3( x 4 - 3 x 2 - 4) . Equivalently, f ( x ) = - 3 x 4 + 9 x 2 + 12. Solution f (3) = a n (3 4 - 3 h 3 2 - 4) = - 150 To find a n , use the fact that f (3) = - 150. a n (81 - 27 - 4) = - 150 Solve for a n . 50 a n = - 150 a n = - 3
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