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 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 cannot be greater than the number of sign changes in f ( − x ) . Calculation: It is given that the polynomial equation is D 5 + D 4 − 9 D 3 − 5 D 2 + 16 D + 12 = 0 . Here, f ( D ) = D 5 + D 4 − 9 D 3 − 5 D 2 + 16 D + 12 and the degree is 50. Move from left to right, obtain the number of sign changes. There are 2 sign changes; hence the number of positive roots is 2. Change the function f ( D ) to f ( − D ) as follows. f ( − D ) = ( − D ) 5 + ( − D ) 4 − 9 ( − D ) 3 − 5 ( − D ) 2 + 16 ( − D ) + 12 = − D 5 + D 4 + 9 D 3 − 5 D 2 − 16 D + 12 As there are 3 sign changes, the number of negative roots is 3. Hence, the maximum possible numbers of positive and negative roots of D 5 + D 4 − 9 D 3 − 5 D 2 + 16 D + 12 = 0 are 2 and 3, respectively. Thus, the possible rational roots by Rational Root Theorem are, r r = integer factor of 12 integer factor of 1 = 1 , 2 , 3 , 4 , 6 , 12 1 = ± 1 , ± 2 , ± 3 , ± 4 , ± 6 , ± 12 Begin with − 1 , perform synthetic division as follows. 1 1 − 9 − 5 16 12 − 1 0 9 − 4 − 12 1 0 − 9 4 12 0 [ − 1 ] The remainder is 0

Basic Technical Mathematics with Calculus (11th Edition)

11th Edition

ISBN: 9780134437736

Author: Allyn J. Washington, Richard Evans

Publisher: PEARSON

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Chapter 15.3, Problem 17E

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Chapter 15.3, Problem 16E

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Chapter 15 Solutions

Basic Technical Mathematics with Calculus (11th Edition)

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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