Showing that a polynomial is surjective

Question

How do I formally show that a polynomial, say $f(x)=x^3 - 5$, is surjective? For example, if $f(x)$ were a linear function such as $f(x) = 5x$, then I would simple need to show that for all $y$ in the range, there is an $x$ such that $f(x)=y$ and I would choose that $x = y/5$. My intuition tells me that it has something do with the fact that polynomials are continuous, i.e. that all of their range of $\mathbb{R}$ is mapped to by $f(x)$. Please guide me as to how to formally prove this for any polynomials of odd degree $\geq 3$.

Answer 1

It is continuous, $\displaystyle\lim_{x\rightarrow +\infty}x^3=+\infty\;,\;\lim_{x\rightarrow -\infty}x^3=-\infty$

There exists $a,b$ such that $f(a)<-N, f(b)>N$, $f([a,b])$ is an interval which contains $[-N,N]$