Fundamental Theorem of Algebra for highschool My teacher has told me about the Fundamental Theorem of Algebra, but I can't seem to find any proofs on it which I can understand. For something so important I'm hoping to find a proof that a highschool student can understand.
References are welcome too, thanks!

There is one intuitive proof for this theorem. Let $p(z)=\sum_{i=0}^na_iz^i$ where $a_i\in \mathbb{C}$. We need to show that $p$ has a zero in $\mathbb{C}$. Take a very large real number $R$ and consider the circle $|z|=R$ in $\mathbb{C}$. Since $R$ is large, plugging these $z$ into $p$, we have that $p(z)$ is some very large contour that is somewhat centered at $0$. (Since $R$ is large, the contribution of $a_n z^n$ dominates, so $p(z)\sim a_nz^n$). Similarly, consider the circle $|z|=r$ for some real number $r$ close to zero, then $p(z)\sim a_0$, hence $p(z)$ is some small contour near $a_0$.
When we let $r$ vary from very small to very large, then we get growing contours in $\mathbb{C}$ whose centers vary from $a_0$ to $0$. At some point some contour must go through the origin.
One can make an argument like this precise, however it is very difficult to do so.
