Factorizing Cubic Equations.

Question

Factorization of Cubic Equations has always obstructed my way to the solution to a problem. Is there any simple technique to factorize them?

Answer 1

Here's one technique that works for finding roots of cubic polynomials with integer coefficients when $p$ has at least one rational root: Any cubic polynomial $$p(x) = ax^3 + bx^2 + cx + d$$ always has at least one real root, and so it admits a factorization $$p(x) = (x - r)(Ax^2 + Bx + C). \qquad (\ast)$$ Now, the Rational Root Theorem says that any rational root of $p$ has the form $r = \frac{m}{n}$ where $m | d$, $n | a$, and $\gcd(m, n) = 1$. In other words, this gives us a recipe for generating a list that contains any rational root of $p$. If there is such a root, we can find it and use polynomial long division to give a factorization $(\ast)$ of $p$. This reduces the problem to finding the roots of the quadratic.

Other methods exist and can be superior for certain classes of polynomials.

If $p$ doesn't have a rational root, finding exact roots is generally much harder. Exhaustive algorithms exist but they are unpleasant and sometimes return simple solutions in a form difficult to recognize.