<s>[INST] 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? 
 [/INST] The graph of a function $y=f(x)$ intersects the $x$-axis when $y=0$, that is when $f(x)=0$.
Now if a factor $(x-a)$ appears in your polynomial, then the polynomial is zero when $x-a = 0$, which occurs precisely when $x=a$. Therefore
$$\begin{align*}
(x-a)(x-b) = 0 & \Leftrightarrow x-a=0 \text{ or } x-b = 0\\
& \Leftrightarrow x=a \text{ or } x=b
\end{align*}$$
So the graph $y=(x-a)(x-b)$ intersects the $x$-axis when $x=a$ or $x=b$.
This explains the $-$ signs.
So the quadratic $(x+3)(x+2)$ has roots $-3$ and $-2$, whereas the quadratic $(x-3)(x-2)$ has roots $3$ and $2$.
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