2021/108) The polynomial: $P(x) = 1 + a_1x +a_2x^2+...+a_{n-1}x^{n-1}+x^n$ with non-negative integer coefficients has $n$ real roots. Prove, that $P(2) \ge 3^{n}$

Because all coefficients are positive so all n roots are -ve and hence

$P(x) = \prod_{k=1}^{n} (x+ a_k)$ where all $a_k$ are positive

Further $\prod_{k=1}^{n} (a_k) = 1$

So $P(2) = \prod_{k=1}^{n}(2+a_k)\cdots(1)$

Now taking AM GM between 1,1 $a_k$ we get $(2+a_k) >= 3\sqrt[3]{(a_k)}\cdots(2)$

So from (1) and (2)

$P(2) >= 3^n \sqrt[3]{\prod_{k=1}^{n} (a_k)}  = 3^n$ and hence $P(2) >= 3^n$