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

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

marked as duplicate by David Richerby, Raphael♦Nov 21 '15 at 11:05

• Take care of adhereing to the exact definition of $\Theta$ you were given in class. Are functions with negative values allowed? – Raphael Nov 21 '15 at 11:08
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}