Cubic Equations
The calculator solves the cubic equation ax³ + bx² + cx + d = 0: rational root theorem, synthetic division, Cardano's formula.
The cubic equation solver solves ax³ + bx² + cx + d = 0. If the equation has a rational root, the calculator finds it with the rational root theorem, divides the polynomial using synthetic division (Horner's method) and solves the remaining quadratic exactly. Otherwise it uses Cardano's formula or the trigonometric method and gives approximate roots.
Method 1. Finding a rational root
If the coefficients are whole numbers, a rational root p/q (in lowest terms) must have p dividing the constant term d and q dividing the leading coefficient a. Try these numbers, find a root x0 and divide the polynomial by (x − x0) — a quadratic remains.
Example. x3 − 6x2 + 11x − 6 = 0
Divisors of 6: ±1, ±2, ±3, ±6. Try x = 1: 1 − 6 + 11 − 6 = 0 — a root.
Synthetic division: x3 − 6x2 + 11x − 6 = (x − 1)(x2 − 5x + 6)
x2 − 5x + 6 = 0 ⇒ x = 2, x = 3. Answer: x = 1; 2; 3
Synthetic division
Dividing a polynomial by (x − x0) is conveniently written as a table: bring the first coefficient down, then each next number = coefficient + x0 · the previous bottom number. The last number is the remainder; if x0 is a root, it is zero.
Method 2. Cardano's formula
The substitution x = t − b/3a turns the equation into t3 + pt + q = 0. The number of roots depends on the sign of
Δ = (q/2)2 + (p/3)3
- Δ > 0 — one real root t = ∛(−q/2 + √Δ) + ∛(−q/2 − √Δ) and two complex ones;
- Δ = 0 — all roots are real and some coincide;
- Δ < 0 — three distinct real roots, found with the trigonometric method.
A cubic equation with real coefficients always has at least one real root.