Recap: Solving Polynomial Equation P=Q in Terms of Factoring P-Q

This is essentially a recap. Recall from here that the solutions of a polynomial equation $P(x)=Q(x)$ (here $P,Q$ are two polynomials) are simply the roots of the single polynomial $P(x)-Q(x)$. Furthermore, by the factor theorem, the roots $a$ of $P(x)-Q(x)$ correspond to factors $x-a$ of the polynomial $P(x)-Q(x)$.

In other words: to find the solutions to $P(x) = Q(x)$, it is enough to find the roots of $P(x)-Q(x)$, which is conceptually closely related to the factorization of $P(x)-Q(x)$.

Throughout this section, we will organize our thoughts in terms of factorization, but everything we do will still connect to the goal of solving polynomial equations and inequalities (both linear and nonlinear)!