Today, I was working on some limit practice problems and came across two that I had to factor.
The first limit had this polynomial in the denominator: $$x^2+2x-15$$
which I factored down to: $$(x-3)(x+5)$$
The second limit had this polynomial in the numerator: $$2z^2-17z+8$$
which I factored down to: $$(2z-1)(z-8)$$
As I was looking over these problems, I realized I don't know why polynomials factor down like this. I was just taught what to do when I come across each type. When I factor them down the answers "make sense", but I just can't see a reason why they do as I look at it from my current perspective. Could someone shed some light on this? Are there proofs for things like this?