Q2014/007) show that the five roots of the quintic

a5 x^5+ a4x^4+a3x^3+a2x^2+a1x+a0=0 are not all real if 2a2^4<5a5 a3.="" nbsp="" p="">

I shall prove it by contradiction.
Without loss of generality let a5 = 1
 Let the roots be y1,y2, y3,y4,y5
 Then a4^2 = (y1+y2+y3+y4+y5)^2 = y1^2 + y2^2 + y3^2 + y4^2 + y5^2 + 2a3 as a3 consists of product of 2 elements that are separate

Or 2a4^2 =4a3 + 2y1^2 + 2y2^2 + 2y3^2 + 2y4^2 + 2y5^2) 

= 4a3 + ½(4y1^2 + 4y2^2 + 4y3^2 + 4y4^2 + 4y5^2) 

= 4a3 + ½(sum(ym-yn)^2+ 2ymyn)

>= 4a3 + a3 as sum (ym-yn)^2 >=0 and sum ymyn= a3

 >= 5a3

So all the roots are real => 2a4^2 >= 5a3

So under given condition all roots cannot be real