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Show that the five roots are not real

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anemone

MHB POTW Director
Staff member
Feb 14, 2012
3,756
Hi MHB,

I have encountered a problem recently for which I couldn't think of even a single method to attempt it, and this usually is an indicator that a problem really isn't up my alley. That notwithstanding, I don't wish yet to concede defeat. Could someone please show me at least some idea on how to crack it? Thanks in advance.

Problem:

Show that the five roots of the quintic $a_5x^5+a_4x^4+a_3x^3+a_2x^2+a_1x+a_0=0$ are not all real if $2a_4^2<5a_5a_3$.
 

kaliprasad

Well-known member
Mar 31, 2013
1,322
Hi MHB,

I have encountered a problem recently for which I couldn't think of even a single method to attempt it, and this usually is an indicator that a problem really isn't up my alley. That notwithstanding, I don't wish yet to concede defeat. Could someone please show me at least some idea on how to crack it? Thanks in advance.

Problem:

Show that the five roots of the quintic $a_5x^5+a_4x^4+a_3x^3+a_2x^2+a_1x+a_0=0$ are not all real if $2a_4^2<5a_5a_3$.
I would prove it by contradiction
Without loss of generality let a5 = 1
Let all roots be real 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 because 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))) m is not n
>= 4a3 + a3 as sum (ym-yn)^2 >=0 and sum ymyn= a3
>= 5a3

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

Or if the condition is not satisfied then all root cannot be real
 
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