Use Descartes' rule of signs to find the number of possible positive,negative,and nonreal roots for the following equation. 4x ⁷ + 9x ² + 5x + 10 = 0 A)0 positive;1 negative;6 nonreal B)2 positive;1 or 3 negative;4 or 0 nonreal C)0 positive;0 negative;7 nonreal D)0 positive;0 or 2 negative;5 or 7 nonreal E)none of these

Question

Quizplus · Accepted Answer

Descartes' rule of signs for the given polynomial $4x^{17} + 9x^{12} + 5x + 10 = 0$ indicates 0 changes in sign for positive roots, suggesting 0 positive roots. For negative roots, substituting $x = -x$ gives $4(-x)^{17} + 9(-x)^{12} + 5(-x) + 10$, which results in 2 sign changes, indicating 2 or 0 negative roots (since the number of negative roots is either the number of sign changes or less than that by an even number). The total number of roots for a 17th degree polynomial is 17, so if there are 0 positive and 2 or 0 negative roots, there must be 17 or 15 nonreal roots, respectively. None of the provided options accurately reflects this analysis.

Use Descartes' Rule of Signs to Find the Number of Possible