Can someone please explain why we are able to factor an $n$ degree polynomial function using only it roots?
What I mean is this:
Lets say we have a function defined like so: $$f(x) = ax^4 + bx^3 +\dots$$ It can supposedly be factored like so: $$f(x) = a(x−p)(x−q)(x−r)\dots$$ Where $p, q, r$ etc. are the solutions of the function being equal to $0$. Is there a simple proof for why this is valid, and where does the coefficient $a$ in the factored form come from? (I don't want some lame answer for $a$ like :"if $a$ wasn't there the factored form wouldn't equal the original form"