Write the cubic in standard form: (ax^3+bx^2+cx+d)
Factor out the greatest common factor first, if any
Use the Rational Root Theorem to list possible rational zeros
Test each possible zero by direct substitution or synthetic division
If (r) is a root, divide the cubic by ((x-r))
Factor the resulting quadratic, if possible
Use methods like factoring by grouping, trinomial factoring, or the quadratic formula for the remaining factor
Check the product of all factors by expanding
If no rational root works, look for special patterns or use numerical/advanced methods
For a monic cubic with one known root (r), write it as ((x-r)(x^2+px+q))
Solve for (p) and (q) by matching coefficients
If the cubic has repeated roots, factor them as repeated linear factors
Final form is typically ((x-r_1)(x-r_2)(x-r_3)) or ((x-r)(ax^2+bx+c))