MHB Show that the five roots are not real

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The discussion revolves around proving that not all roots of the quintic equation $a_5x^5+a_4x^4+a_3x^3+a_2x^2+a_1x+a_0=0$ are real under the condition $2a_4^2<5a_5a_3$. A proposed method involves proof by contradiction, assuming all roots are real and deriving inequalities from the relationships between the coefficients and the roots. The argument shows that if all roots were real, it would lead to a contradiction with the given condition. Ultimately, this indicates that at least one root must be non-real if the inequality holds. The discussion emphasizes the mathematical reasoning behind this conclusion.
anemone
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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 $2a_4^2<5a_5a_3$.
 
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anemone said:
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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