Prove that polynomial is always greater than zero Let $p(x)$ be a polynomial such that $p(1) = 0$ and $p'(x) &gt; p(x)$ for all $x \geq 1$ , then we have to prove $p(x) &gt; 0$, for all $x &gt;1$. 

I got it when I thought it logically . But how can we prove it mathematically.

According to me $p'(1)$ should be positive .

So after $ x= 1$ the polynomial will be positive and hence the $p'(x)$ cannot be negative i.e the polynomial is decreasing as polynomial is positive. Hence polynomial is always positive.

But how can we prove it mathematically that the polynomial is always positive . I am not getting any idea  .
Consider a function $h(x) = p(x) e^{-x} ,$ we have $h' (x) =e^{-x} (p'(x) -p(x) )&gt;0 $ for $x&gt;1$ thus $h$ is increasing and therefore $h(x)&gt;h(1) $ for $x&gt;1.$
I have done at last . I don't know which all polynomial are satisfying , but I have proved it .