MHB Polynomial inequality

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The polynomial \( P(x) = 1 + a_1x + a_2x^2 + ... + a_{n-1}x^{n-1} + x^n \) with non-negative integer coefficients is discussed in relation to its real roots and the inequality \( P(2) \ge 3^n \). There is a consensus that the coefficients do not necessarily need to be integers for the inequality to hold, as the constant term and the leading coefficient are both 1. Participants clarify the conditions under which the polynomial has \( n \) real roots and the implications for the inequality. The discussion emphasizes the importance of the polynomial's structure in proving the stated inequality. Overall, the focus remains on establishing the validity of the inequality under the given conditions.
lfdahl
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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 $P(2) \ge 3^{n}$
 
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lfdahl said:
The polynomial: $P(x) = 1 + a_1x +a_2x^2+...+a_{n-1}x^{n-1}+x^n$

with non-negative coefficients has $n$ real roots. Prove, that $P(2) \ge 3^{n}$
I guess that should be "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." Otherwise the result is not true.
 
Hint:

Consider the sign of the roots ... - and then take a closer look at $P(2)$
 
lfdahl said:
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 $P(2) \ge 3^{n}$

I do not see why Opalg mentioned that coefficients should be integers
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 $P(2) >= 3^n$
 
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kaliprasad said:
I do not see why Opalg mentioned that coefficients should be integers
You are quite right. I somehow overlooked the fact that the constant term and the coefficient of $x^n$ are both $1$.
 
kaliprasad said:
I do not see why Opalg mentioned that coefficients should be integers
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)
$P(2) >= 3^n \sqrt[]{\prod_{k=1}^{n} (a_k)} = 3^n$ and hence $P(2) >= 3^n$

Thankyou, kaliprasad, for your solution. Well done!
 
kaliprasad said:
I do not see why Opalg mentioned that coefficients should be integers
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)
$P(2)>=3^n\sqrt[]{\prod_{k=1}^n (a_k)}=3^n$ and hence $P(2)>=3^n---(*)$
typo
(*)should be
$P(2) \geq 3^n \sqrt[3]{\prod_{k=1}^{n} (a_k)} = 3^n$ and hence $P(2) \geq 3^n$
 
Albert said:
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)
$P(2)>=3^n\sqrt[]{\prod_{k=1}^n (a_k)}=3^n$ and hence $P(2)>=3^n---(*)$
typo
(*)should be
$P(2) \geq 3^n \sqrt[3]{\prod_{k=1}^{n} (a_k)} = 3^n$ and hence $P(2) \geq 3^n$

I have done the needful
 

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