I am currently learning how to find zeroes of polynomials. However, 4 things confuse me.
The first question I have is why exactly does the rational zeroes theorem hold true? I get how to use it, but I don't get why it is true. Can someone please explain this?
The second question I have is why is the Descartes rule of signs true?
My third question is how do we find the imaginary roots of polynomial? It can't be through the rational zeroes theorem! But if we don't need to find imaginary roots, why else would we need things like the conjugate pairs theorem?
My last question is of synthetic division in finding the zeroes of polynomials. I don't get why using synthetic division to test if a root is a zero of a polynomial is by taking it, and synthetically dividing it. What I don't get is, why don't you need to change the sign of the zero before synthetically dividing here? Let's say I suspected that the zero of a polynomial is x=-4. Don't I need to change the sign of the root to x+4=0, then synthetically divide by 4? Why is that you don't need to do this?
Can you explain all of this using simple algebra, without complicated techniques? I don't understand any complicated techniques and theorems beyond the quadratic formula. Can you also show and explain your working, so it is easier for me to follow through? I am still a beginner, so that would help very much.