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 4 x 3 − 16 x 2 + 21 x − 9 = 0 . Here, f ( x ) = 4 x 3 − 16 x 2 + 21 x − 9 and the degree of the polynomial is 3. Move from left to right, obtain the number of sign changes. From 4 x 3 → − 16 x 2 , − 16 x 2 → + 21 x , + 21 x → − 9 , there are 3 sign changes. Thus, the number of positive roots is 3. Change the function f ( x ) to f ( − x ) as follows. f ( − x ) = 4 ( − x ) 3 − 16 ( − x ) 2 + 21 ( − x ) − 9 = − 4 x 3 − 16 x 2 − 21 x − 9 As there is no sign change, the number of negative roots is 0. Hence, the maximum possible numbers of positive and negative roots of 4 x 3 − 16 x 2 + 21 x − 9 = 0 are 3 and 0, respectively

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 10E

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Chapter 15.3, Problem 9E

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

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