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Problem: Let $x^3 + ax^2 + bx + c$ be a polynomial with real nonnegative roots. Prove that b>= 3[(c^(2/3)]

 

I turned the equation into b/3 >= cuberoot(c^2) so that it's clearly a use of AM GM, but I'm not sure where to go from here.

Any help or a small hint would be appreciated, thanks :)

 Jun 24, 2021
 #1
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Your idea of using the inequality on AM and GM is correct.

Let $\alpha, \beta, \gamma$ be the real nonnegative roots of the given polynomial,
thus, $x^3+ax^2+bx+c = (x-\alpha)(x-\beta)(x-\gamma)$.  By Vieta's formula,
$\alpha\beta+\beta\gamma+\gamma\alpha = b$ and $\alpha\beta\gamma=c$.

Can you do the rest?

 Jun 24, 2021
 #2
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+1

Ohhh thank you so much, so the three terms that were used in the Am Gm thing are the 3 things that sum to b, Thanks so much :)

Guest Jun 24, 2021

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