Polynomial tending to infinity Take any polynomial $(x-a_1)(x-a_2)\ldots(x-a_n)$ with roots $a_1, a_2,\ldots,a_n$  where we order them so that $a_{i+1}>a_i$ is increasing so $a_n$ is the biggest root. It doesn't matter whether the polynomial is of odd or even degree or whether there is a minus sign in front of highest power of $x$ since all these cases can be treated similarly. How do you show rigorously that after $x=a_n$ the polynomial is a strictly increasing function?

It is the product of positive and strictly increasing functions.
Alternatively: The derivative of $p$ has degree $n-1$ and at least one root in each interval $(a_i,a_{i+1})$ (Rolle). Hence it has no root besides these, especially it does not change signs beyond $x=a_n$.
