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5.5 RealZeros - Professor P Bishop(MDN MAC1147 R.6...

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Professor P. Bishop MAC1147 (MDN) R.6 Synthetic Division (Condensed) Synthetic division is used to divide a polynomial of degree 1 or higher by a binomial g(x ) of the form x – c . Use synthetic division to determine whether x – c is a factor of f(x): 1. ( 29 ( 29 = + - + - = + f x x x x x ; g x x 4 3 2 8 1 5 4 4 2. ( 29 ( 29 = + - = - f x x x ; g x x 2 3 5 4 5.5 The Real Zeros of a Polynomial Function Remainder Theorem: If f(x) is divided by x-c, then the remainder is f(c). Factor theorem: If f is a polynomial function, then x – c is a factor of f(x) if any only if f(c) = 0 . Number of Real Zeros Theorem: A polynomial function cannot have more real zeros than its degree. Descartes’ Rule of Signs: If f is a polynomial function in standard form, then (a) the number of positive real zeros of f either equals the number of variations in the sign of the nonzero coefficients of f(x) or else equals that number less an even integer; (b) ) the number of negative real zeros of f either equals the number of variations in the sign of the nonzero coefficients of f(-x)
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