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

This question already has an answer here:

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

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

• What have you tried and where did you get stuck? Without a specific question, this is just a duplicate of our reference question which explains general strategies of proving such claims. – Raphael Nov 21 '15 at 11:06
• 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

## 1 Answer

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.