My question is very basic, as I do not understand the concept of rewriting a (complex) polynomial into a product of terms using the roots of the polynomial.
I have encountered the fundamental theorem of algebra many times, but never in a concise way. It either stated that a polynomial contains at least one root/a polynomial of degree d has at most d roots/a polynomial can be rewritten into a product. The last one, even though all statements are nonequivalent, interests me the most.
I have seen many proofs relying on the fact that if you solve the roots of any given polynomial (including imaginary roots if present), you can rewrite it into a product with factors $(x-z)$ if $x=z$ is a root. So for instance, a very accessible example would be rewriting $x^2-3x+2=(x-2)(x-1)$. But when higher powers come into the game or even an infinite series (Euler's proof of Basel's problem) I can impossibly work the polynomials out in order to check if it is true.
How could you prove to me in reasonable elementary terms that a polynomial can be written in its "summation" form as well as its "product" form using the roots? When I look up proof of fundamental theorem of algebra, they seem to focus on proving the presence of roots rather than this rewriting. I do not even know for sure if this is regarded as the fundamental theorem of algebra, maybe it is something completely different from what I think it is.
Please help me out in here!
EDIT: Additional question: why do we use the roots, that is to say when $P(x)=0$, to rewrite and not solutions for $P(x)=1$ or $P(x)=-2$ for instance? Why is the root specially suited for this?