Quote:
Originally Posted by CaptainBlack  Let the polynomial:
f(x) = x^n + a_1 x^{n-1} + ... + a_{n-1} x +1
with non-negative real coefficients a_1, .. , a_{n-1} have n real roots.
Prove that f(2) >= 3^n
RonL |
Prove inductively:
First, show it is true for the first term: n = 1
f(2) = 2^(1) + 1 = 3 >= 3^(1) = 3
Now, assume it is true for some value: n = k
f(2) = 2^k + a_1*2^(k-1) + ... + a_{k-1}*2 + 1 >= 3^k
Show that it is true for the next term: n = k + 1
f(2) = 2^(k+1) + a_1*2^k + ... + a_{k}*2 + 1 >= 3^(k+1)
We have:
2^(k+1) + a_1*2^k + ... + a_{k}*2 + 1
= 2*[2^k + a_1*2^(k-1) + ... + a_{k-1}*2 + a_{k}] + 1
It just occurred to me that I can go no futher until I know how to use the fact that there are n real roots. I know that is important to the proof (because it limits the values of a_1 to a_n), so I need to figure out how that effects the problem before I can go on.