Explanation Theorem used: Rational Root Theorem: Any rational root r r of a polynomial equation f ( x ) = a n x n + a n − 1 x n − 1 + ⋯ + a 0 = 0 is an integer factor of a 0 divided by an integer factor of a n . Therefore, r r = integer factor of a 0 integer factor of a n . Definition used: Descartes’ Rule of Signs: The number of positive zeros of a polynomial equation f ( x ) = 0 cannot be greater than the number of changes in sign in f ( x ) in moving from one term to the next in f ( x ) . The number of negative zeros cannot be greater than the number of sign changes in f ( − x ) . Calculation: It is given that the polynomial equation is 8 x 4 − 32 x 3 − x + 4 = 0 . Here, f ( x ) = 8 x 4 − 32 x 3 − x + 4 and the degree is 4. Move from left to right, obtain the number of sign changes. From 8 x 4 → − 32 x 3 and − x → + 4 , there are 2 sign changes. Thus, the number of positive roots is 2. Change the function f ( x ) to f ( − x ) as follows. f ( − x ) = 8 ( − x ) 4 − 32 ( − x ) 3 − ( − x ) + 4 = 8 x 4 + 32 x 3 + x + 4 As there are no sign changes, the number of negative roots is 0. Hence, the maximum possible numbers of positive and negative roots of 8 x 4 − 32 x 3 − x + 4 = 0 are 2 and 0, respectively. Thus, the possible rational roots by Rational Root Theorem are, r r = integer factor of 4 integer factor of 8 = 1 , 2 , 4 1 , 2 , 4 , 8 = ± 1 , ± 2 , ± 4 , ± 1 2 , ± 1 4 , ± 1 8 Begin with 4, perform synthetic division as follows

Mylab Math With Pearson Etext -- 18 Week Standalone Access Card -- For Basic Technical Mathematics With Calculus

11th Edition

ISBN: 9780135902912

Author: Allyn J. Washington

Publisher: PEARSON

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Chapter 15.3, Problem 12E

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Chapter 15.3, Problem 11E

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Mylab Math With Pearson Etext -- 18 Week Standalone Access Card -- For Basic Technical Mathematics With Calculus

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