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 5 n 4 − 2 n 3 + 40 n − 16 = 0 . Here, f ( x ) = 5 n 4 − 2 n 3 + 40 n − 16 and the degree is 4. Move from left to right, obtain the number of sign changes. From 5 n 4 → − 2 n 3 , − 2 n 3 → + 40 n and + 40 n → − 16 , there are 3 sign changes. Thus, the number of positive roots is 3. Change the function f ( n ) to f ( − n ) as follows. f ( − n ) = 5 ( − n ) 4 − 2 ( − n ) 3 + 40 ( − n ) − 16 = 5 n 4 + 2 n 3 − 40 n − 16 There is one sign change, thus the number of negative roots is 1. Hence, the maximum possible numbers of positive and negative roots of 5 n 4 − 2 n 3 + 40 n − 16 = 0 are 3 and 1, respectively. Thus, the possible rational roots by Rational Root Theorem are, r r = integer factor of 16 integer factor of 5 = 1 , 2 , 4 , 816 1 , 5 = ± 1 , ± 2 , ± 4 , ± 8 , ± 16 , ± 1 5 , ± 2 5 , ± 4 5 , ± 8 5 , ± 16 5 Factorize the left hand side, obtain the value of n

Student Solutions Manual For Basic Technical Mathematics And Basic Technical Mathematics With Calculus

11th Edition

ISBN: 9780134434636

Author: Allyn J. Washington, Richard Evans

Publisher: PEARSON

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Chapter 15.3, Problem 13E

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Chapter 15.3, Problem 12E

Chapter 15.3, Problem 14E

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Chapter 15 Solutions

Student Solutions Manual For Basic Technical Mathematics And Basic Technical Mathematics With Calculus

Ch. 15.1 - Use the factor theorem to determine whether t − 2...Ch. 15.1 - Prob. 2PE Ch. 15.1 - Prob. 3PE Ch. 15.1 - Prob. 4PE Ch. 15.1 - Prob. 1E Ch. 15.1 - Prob. 2E Ch. 15.1 - In Exercises 1–4, make the given changes in the...Ch. 15.1 - Prob. 4E Ch. 15.1 - Prob. 5E Ch. 15.1 - Prob. 6E

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