How to prove that a polynomial of degree n is θ(x^n)

Question

How can I prove that if $T(x)$ is a polynomial of degree $n$ then $T(x) = \Theta(x^n)$.

Answer 1

Say $T(x) = a_n x^n + \dotsm + a_0$, then by the triangle inequality for $x \ge 1$:

$\begin{align} \lvert T(x) \rvert &\le \lvert a_n \rvert x^n + \lvert a_{n - 1} \rvert x^{n - 1} + \dotsm + \lvert a_0 \rvert \\ &\le \lvert a_n \rvert x^n + \lvert a_{n - 1} \rvert x^n + \dotsm + \lvert a_0 \rvert x^n \\ &= (\lvert a_n \rvert + \lvert a_{n - 1} \rvert + \dotsm + \lvert a_0 \rvert) x^n \end{align}$

I'm sure you can take it from here. You'll have to figure out an appropriate lower bound to match.