Answer the question.
-If $\mathrm { n }$ is an odd number, then which of the following statements is true?
A) The equation $x ^ { n } + 1 = 0$ has an even number of real roots.
B) The equation $x ^ { n } + 1 = 0$ has $n + 1$ roots.
C) The equation $x ^ { n } + 1 = 0$ has an odd number of nonreal roots.
D) The equation $\mathrm { x } ^ { \mathrm { n } } + 1 = 0$ has exactly one real root.

Question

Quizplus · Accepted Answer

For an odd $n$, the equation $x^n + 1 = 0$ can be rewritten as $x^n = -1$. Since $n$ is odd, the function $f(x) = x^n$ is continuous and changes sign from negative to positive (or vice versa) as $x$ crosses zero, ensuring exactly one real root where the function equals -1. The other roots, if any, must be complex and come in conjugate pairs, but the question of their number or existence isn't addressed by this specific choice.

Answer the Question n\mathrm { n }n Is an Odd Number, Then Which of the Following Statements

Answer the Question $\mathrm { n }$ Is an Odd Number, Then Which of the Following Statements