My current level of maths does not allow me to understand any of the proofs I was able to find online for the Fundamental Theorem of Algebra. I find it very unsettling having to continue learning about polynomials without being able to grasp such a fundamental property of them that is there exists a complex root for any polynomial with complex coefficients.
My question is, is there a way I can reach the same conclusion of the Fundamental Theorem of Algebra for polynomials with real coefficients instead of complex coefficients. In other words, to prove that at least one complex root exists for a polynomial with real coefficients. Does restricting the coefficients to be real instead of complex make things any easier?
If the answer to my question above is no, then I would appreciate it if you could provide any sources that I could've missed, that explain the Theorem by using the least amount of mathematical notation and advanced concepts as possible.
If I cannot figure this out, I am going to have to accept the existence of a root as an axiom going further, which is something I really don't want to do.
Best Answer
The fundamental theorem of algebra is exactly as difficult for real vs. complex coefficients. The reason is that if $f(x) = f_0 + \dots + f_n x^n$ is any polynomial with complex coefficients, then the polynomial
$$\overline{f(\overline{x})} f(x) = (\overline{f_0} + \dots + \overline{f_n} x^n)(f_0 + \dots + f_n x^n)$$
has real coefficients (because it is invariant under coefficient-wise complex conjugation), and its roots are exactly the roots of $f$ together with their complex conjugates (this is a nice exercise). So any proof of the FTA for real coefficients immediately yields the FTA for complex coefficients.
It speaks well of you that you want to understand this result instead of taking it on faith; unfortunately it is one of the first "genuinely difficult" results one comes across in mathematics. There are no really easy proofs, one has to really understand something. You can find a lovely collection of proofs here on MO but all their prerequisites involve a certain amount of analysis or topology or in one case Galois theory and this is unavoidable. (The Galois theory is used to reduce the amount of analysis necessary to the fact that a real polynomial of odd degree has a real root (edit: and, as Daniel Schepler points out, to the fact that a non-negative real number has a real square root), which follows from the intermediate value theorem.)
FWIW I think this proof shared by Kevin McGerty is the least technical on the list but it still requires some familiarity with analysis.