Cubics look intimidating, but most exam cubics are designed to have one easy root. Find it, divide it out, and you're left with a quadratic you already know how to solve — and a cubic equation solver confirms all three roots at once.
On this page
How many roots?
A cubic always has three roots (counting multiplicity). Since complex roots arrive in conjugate pairs, there's always at least one real root — the foothold for our method.
The find-one-then-factor strategy
- Find a rational root by the rational root theorem: test factors of the constant ÷ factors of the leading coefficient.
- Divide it out to reduce the cubic to a quadratic.
- Solve the quadratic by any method you like.
Worked example
Solve x³ − 6x² + 11x − 6 = 0.
- Candidates: ±1, ±2, ±3, ±6. Test x = 1: 1 − 6 + 11 − 6 = 0 ✔, so (x − 1) is a factor.
- Dividing out gives x² − 5x + 6.
- Factor: (x − 2)(x − 3).
Roots: x = 1, 2, 3.
When two roots are complex
Sometimes the leftover quadratic has a negative discriminant.
Factor as (x − 1)(x² + x + 1). The quadratic gives x = (−1 ± √−3)/2 = −0.5 ± 0.866i. So the three roots are 1 and a complex-conjugate pair.
To interpret those complex roots, see the complex number calculator guide.
Checking on the solver
Switch to EQN mode, pick a degree-3 polynomial, and enter a, b, c, d:
1, −6, 11, −6 → x = 1, 2, 3The equation solver pillar guide covers the rest of EQN mode.
Frequently asked questions
How many roots does a cubic have?
Three, counting multiplicity, with always at least one real root.
How do I solve without the cubic formula?
Find one rational root, divide it out, then solve the remaining quadratic.
What is the rational root theorem?
Rational roots p/q have p dividing the constant and q the leading coefficient — a short candidate list.
Solve your cubic
Enter the four coefficients in EQN mode and read all three roots.
Open the cubic equation solver →