TITLE: Is it possible to prove the Fundamental Theorem of Algebra for all polynomials of degree $n \le 4$?
QUESTION [6 upvotes]: Recenly I've been wondering whether it's possible to prove the FTA for all polynomials of degree $n \le 4$ without utilizing advanced maths but, at most, basic linear algebra (concepts such as eigenvectors, eigenvalues, determinants etc.). I've tried to give this some thought but I've been only able to make very naïve and futile statements such as "all real numbers can be represented as complex numbers " and "some quadratics and quartics, such as $p(x)=x^2 + 1$ and $p(x)=x^4 -1$, contain complex solutions". Now, I would be really greatful, if it is indeed possible to prove the FTA for all polynomials with degree $n \le 4$, if someone could provide a reference of some sort.

REPLY [4 votes]: Let $p\in\mathbb{R}[X]\setminus\mathbb{R}$ be a polynomial of degree at most $4$.

If $\deg(p)=1$, then you can easily explicit the root of $p$.
If $\deg(p)=2$, same remark.
If $\deg(p)=3$, you can also explicit the roots of $p$ using Cardano's method.
If $\deg(p)=4$, you can also explicit the roots of $p$ using this time Ferrari's method.

Note that if $\deg(p)\geqslant 5$, in the general case, you won't be able to find such formulas for the roots of $p$, it's the Ruffini-Abel theorem