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> (a) Show that the roots of equation $z^4 + a\_1 z^3 + a\_2 z^2 + a\_3 z + a\_4 = 0$ where $a\_1, a\_2, a\_3, a\_4$ are real constants different from zero, has a pure imaginary root if $a\_3^2 + a\_1^2 a\_4 = a\_1 a\_2a\_3$  
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>  (b) Is converse of (a) true?
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I tried factoring $z^4 + a\_1 z^3 + a\_2 z^2 + a\_3 z + a\_4 = (z^2 + b\_1 z + b\_2)(z^2 + c\_1 z + c\_2)$ and compared (equating both $b\_1 = c\_1 = 0$) but couldn't get the required results (that would get $a\_1 = 0 $). Did I make any mistake on assumptions? I am beginning to doubt this question? Any suggestions, hints?

![enter image description here](https://i.stack.imgur.com/1r6X2.jpg)