algebra precalculus - Factorising equation with power 3

When you have polynomial for example $x^2 + 10 x + 25$ and you are asked to factorise, I know that two numbers that multiply to make $25$ and add to make $10$ are $5$ and $5$ and you take out the $x$.

So it becomes $(x + 5) (x + 5)$.

However when you have a polynomial like $x^3 - 4x^2 + 3x$ what is a rule you can use for this ? I know that this becomes $x(x-1) (x-3) = 0$ to find $x$ but I would not know how to do this and would put in unnecessary effort. So is there a rule to factorise these types of polynomials ?

Answer

In general, the answer is yes, but extremely complicated. For most problems you'd face, the rational roots theorem should suffice if you can't see a clear way to factor.

In this case, we see $x=0$ is a possible root, and it is, thus

$$x^3 - 4 x^2 + 3 x = x(x^2 - 4 x + 3)$$

And the rest is a quadratic.