I'm working on a self-made problem where I have a cubic equation I need to factor. I have some flexibility in choosing my own coefficients for the polynomial, so I am strategically trying to choose my coefficients within the admissible set so that the cubic I end up with is simple to factor. What are the forms of some simple cubic equations to factor? For example, if the second coefficient is zero, so it is
$$ax^3 + cx + d = 0$$ is that simpler to factor than the general case? Any other examples? I'm not trying to solve for the general cubic, which is hard, I want to modify my cubic so that it is easy to factor. Thank you!
For the cubic polynomial $ax^3+c x+d$ to factorize, its coefficients $a$, $c$ and $d$, assuming all rational, have to satisfy $$d=-r(ar^2+c)$$ where $r$ is rational. Then, the polynomial factorizes as $$ax^3+c x +d=a(x-r)(x^2+ r x +r^2 +\frac ca)$$