Explanation Given information: All the real roots of a polynomial are negative and the real parts of all of the complex roots are negative. Proof: By using induction to prove the result, Consider a polynomial P n ( x ) = x n + a n − 1 x n − 1 + . . . + a 1 x + a 0 . For n = 1 , P 1 ( x ) = x + a is the first degree polynomial. P 1 ( x ) = x + a has exactly one real root x = − a If the real roots of a polynomial is negative, then − a < 0 ⇒ a > 0 This implies the coefficients of the polynomial are positive. For n = n , P n ( x ) = x n + a n − 1 x n − 1 + . . . + a 1 x + a 0 be n t h degree polynomial with n roots. Say c 1 , c 2 , . . . , c n with Re ( c i ) < 0 for i = 1 , 2 , . . . , n and the real parts of all of the complex roots are negative so a 0 , a 1 , . . . a n − 1 > 0 . Now, to prove the result is true for n = n + 1 P n + 1 ( x ) = x n + 1 + a n x n + a n − 1 x n − 1 + . . . + a 1 x + a 0 be ( n + 1 ) t h degree polynomial with ( n + 1 ) roots. Say c 1 , c 2 , . . . , c n , c n + 1 . Assume that Re ( c i ) < 0 for i = 1 , 2 , . . . , n + 1 to prove that a 0 , a 1 , . . . , a n + 1 > 0 c 1 is one of the root of P n + 1 ( x ) implies that, P n + 1 ( x ) = ( x − c 1 ) ( x n + b n − 1 x n − 1 + . . . + b 1 x + b 0 ) Take q n ( x ) = x n + b n − 1 x n − 1 +

Differential Equations: An Introduction to Modern Methods and Applications

3rd Edition

ISBN: 9781118531778

Author: James R. Brannan, William E. Boyce

Publisher: WILEY

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Question

Chapter 5.9, Problem 4P

To determine

To prove: If all the real roots of a polynomial are negative and the real parts of all of the complex roots are negative, then the coefficients of the polynomial are necessarily positive.

(Thus, if any coefficients are negative or zero, the polynomial must have roots with non-negative real parts. Thisprovides a useful initial check on whether there may be rootsin the right half plane.)

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Chapter 5.9, Problem 3P

Chapter 5.9, Problem 5P

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Chapter 5 Solutions

Differential Equations: An Introduction to Modern Methods and Applications

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Solution Summary: The author explains that if all the real roots of a polynomial are negative, then the coefficients are necessarily positive.

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Chapter 5 Solutions