1how did u come up writing P(x) that way?
$Let $p(x)=x^n+a_{1}x^{n-1}+...............+a_{n-1}x+4$ be a polynomial with\\\\ non-negative coefficient has $n$ real roots.Then prove that $p(1)>2^{n+1}$
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5 Answers
21clearly all roots are negative
so for positive integers xi, i = 1,2,3 .....n
P(x) = (x+a1)(x+a2)....(x+an)
P(1) = (1+a1)(1+a2) ...(1+an)
P(1)≥2n+1 from AM-GM
1i m still unable to get the way u applies am-gm!plz show that last step of applying am-gm[sorry for being dumb]
21No , actually i should have shown it
(1+ a1)≥2√a1 (by AM - GM)
(1+ a2)≥2√a2
.......
(1+an) ≥2√an
Multiplying all of them
we get p(1) ≥ 2n√(a1a2.....an)= 2n+1
because (a1a2.....an) = 4
