Use Descartes's rule of signs to obtain information regarding the roots of the equation. $x ^ { 9 } + 5 = 0$
A) 1 negative root, 8 complex roots
B) 9 complex roots
C) 1 negative root, 9 complex roots
D) 1 positive root, 1 negative real root, 7 complex roots
E) 1 positive root, 8 complex roots

Question

Quizplus · Accepted Answer

The equation $x^9 + 5 = 0$ can be rewritten as $x^9 = -5$, which indicates there is one real negative root (since raising a negative number to an odd power results in a negative number) and the rest of the roots (if any) would be complex. However, since it's a 9th degree polynomial, it must have 9 roots in total (counting multiplicities), so the remaining 8 roots are complex. There are no sign changes in the polynomial itself, indicating no positive roots.

Use Descartes's Rule of Signs to Obtain Information Regarding the Roots