Question

For a∈R, let f(z)=z^5-5z+a. Select the correct statement for α satisfying f(α)=0.

(a) α has exactly three possible real values for a>4

(b) α has exactly one possible real value for a>4

(c) α has exactly three possible real values for a<-4

(d) α has exactly one possible real value for -4<a<4

This question was posed to me during an online interview.

My question comes from Functions of a Complex Variable in chapter Complex Function Theory of Engineering Mathematics

Answer 1

Correct answer is (b) α has exactly one possible real value for a>4

To explain: z^5-5z+a=0 ⇒ z^5-5z=-a ⇒ z(z-5^1/4)(z+5^1/4)(z^2+5^1/2)=-a f'(z)=5z^4–5=0 ⇒ (z^2+1)(z^2-1)=0 ⇒ (z-1)(z+1)(z^2+1)=0 ⇒ α has exactly one possible real value for a>4 and exactly three possible real values for -4<a<4.