factoring polynomials with 3rd degree or higher

Question

I was searching for a method that would allow me to factor polynomials like this one

$x^3 - 13x^2 +(14+4y)x + 8y=0$

I failed, I've only found how to factor by grouping or long division with already given root. That's useless for me in this case, what can I do here?

Answer 1

• Note that the polynomial nay always not be factorizable (Irreducible Factorization) or may involve complex (not real) roots.
• If it is possible, there may be the case:
  • If it is a cubic equation in x like the following, and if it has a rational root of the form $p/q$ then p must divide $a_3$ and q must divide $a_0$ $$a_0x^3+a_1x^2+a_2x+a_3=0$$
  • If you are really serious, you may like to see Cardano's method.
  • If you have many variables, I don't think there is a general method except noticing some pattern or some prior information.

Answer 2

If you rearrange as $$y=-\frac{x^3-13x^2+14x}{4x+8}$$ you will see that there is precisely one real value of $y$ for each real value of $x$ except in the case $x=-2$ when there is no solution.

You can easily factorise numerator and denominator to identify zeros and asymptotes.