Solving of Equation p(x)=0 by Factoring Its Left Side

The essence of method of factoring is following. Suppose we need to solve equation p(x)=0, where p(x) is polynomial of n-th degree. Suppose we factored it as `p(x)=p_1(x)*p_2(x)*p_3(x)`, where `p_1(x),p_2(x),p_3(x)` are polynomials of lesser degree than n. Then, instead of solving equation p(x)=0, we need to solve set of equations `p_1(x)=0,p_2(x)=0,p_3(x)=0`. All found roots of these equations, and only they, will be roots of the equation p(x)=0.

Example 1. Solve equation `x^3+2x^2+3x+6=0`.

Let's factor left side of equation. We have that `x^3+2x^2+3x+6=x^2(x+2)+3(x+2)=(x^2+3)(x+2)`.

Thus, either `x^2+3=0` or `x+2=0`. First equation doesn't have roots, second equation has root x=-2. Therefore, initial equation has one root: x=-2.

Method of factoring is applicable to any equations of the form p(x)=0, where p(x) is not ony polynomial.

Example 2. Solve equation `x^2sqrt(x)-9sqrt(x)=0`.

Domain of this equation is `x>=0`.

After factoring we have that `sqrt(x)(x^2-9)=0`, thus, either `sqrt(x)=0`, or `x^2-9=0`. From first equation x=0, from second equation x=3 and x=-3, but x=-3 is not in the domain of the initial equation.

Therefore, initial equation has two roots: x=0 and x=3.