Question

Let f(z)=z^4+a1z^3+a2z^2+a3z+a4=0; a1, a2, a3, a4 being real and non-zero. If f has a purely imaginary root, then what is the value of the expression a3/(a1a2)+ a1a4/(a2a3) ?

(a) 0

(b) 1

(c) -2

(d) 2

The question was posed to me in an online quiz.

The doubt is from Functions of a Complex Variable topic in portion Complex Function Theory of Engineering Mathematics

Answer 1

Correct answer is (b) 1

The best I can explain: For real x(≠0), let ix be the root⇒x^4-a1x^3i- a2x^2+a3xi+a4=0⇒x^4-a2x^2+a4=0 and a1x^3-a3x=0

a1x^3-a3x=0 ⇒ a1x^2-a3=0 ⇒x^2=a3/a1, putting this value in the equation, a3/(a1a2)+a1a4/(a2a3)=1.