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 21 t 3 + 56 t 2 − 7 = 0 . Reduce the function to 3 t 3 + 8 t 2 − 1 = 0 Here, f ( t ) = 3 t 3 + 8 t 2 − 1 and the degree of the polynomial is 3. Move from left to right, obtain the number of sign changes. From + 8 t 2 → − 1 , there is only 1 sign change. Thus, the number of positive roots is 1. Change the function f ( t ) to f ( − t ) as follows. f ( − t ) = 3 ( − t ) 3 + 8 ( − t ) 2 − 1 = − 3 t 3 + 8 t 2 − 1 From − 3 t 3 → + 8 t 2 and + 8 t 2 → − 1 , there are 2 sign changes. Thus, the number of negative roots is 2. Hence, the maximum possible numbers of positive and negative roots of 3 t 3 + 8 t 2 − 1 = 0 are 1 and 2, respectively. Thus, the possible rational roots by Rational Root Theorem are as follows. r r = integer factor of 1 integer factor of 3 = 1 1 , 3 = ± 1 , ± 1 3 Begin with 1, perform synthetic division as follows

11th Edition

ISBN: 9780134508290

Author: Evans

Publisher: PEARSON CUSTOM PUB.(CONSIGNMENT)

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Chapter 15 Solutions

EBK BASIC TECHNICAL MATHEMATICS

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