Polynomial

Given that the equation x^4+px^3+qx^2+rx+s=0 has four real, positive roots, prove that-
(a)pr-16s\geq 0
(a)q^2-36s\geq 0

Is there any proof without using Cauchy-Schwarz?

3 Answers

1)prâ‰¥ 16s

iff (a+b+c+d)(abc+ bcd+ cda + adb)â‰¥ 16(abcd)

by AM-GM

a+b+c+d â‰¥ 4(abcd)^(1/4)

aind abc+ bcd+ cda + adb â‰¥ 4(abcd)^(3/4)

multiply them..so its true..qed

2) q²â‰¥ 36s

iff (ab+bc+ cd+ ad)² â‰¥ 36abcd

By AM-GM ab+ bc+ cd+ ad â‰¥ 4(abcd)^(1/2)

square it down.. so you probably meant 16 and not 36

or i have missed something?

No, for the second one its 36 only.
You've done a minor mistake.
It should be (ab+bc+cd+ad+ac+bd)² â‰¥ 36abcd

yeah! it is ok.

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