Q:

What do polynomials solve for? Roots?

I had a question in which I’ve been hung up over on. I understand if we had a graph x-y plane and we drew points that intercept the $x$-axis at $2$ and $3$, we would write a quadratic equation that satisfies our condition as $(x+3)(x+2)$, or $x^2+5x+6$, and set this equal to $0$ using the null factor law. Then, we would solve it by factoring and would eventually get back to $(x+3)(x+2)=0$.
And we would get $x=-2,-3$ but how come these are the roots(points that intercept the $x$-axis), when we first created this quadratic equation using the points $2$ and $3$ on $x$-axis? I want to understand in short a logical explanation what we are solving for when we are factoring polynomials? 

A:

The roots of the polynomial $x^2-5x+6$ correspond to the values of $x$ where the function $f(x)=(x-3)(x-2)$ is equal to $0$.
They are points of the $x$-axis : $(3,0)$ and $(2,0)$, i.e. the intersection points of the graph of $f(x)$ with the $x$-axis.

