Roots of cubics

(1) A cubic polynomial f(z) with complex coefficients can have 1, 2, or 3 complex roots.

Counting multiplicities, f(z) always has three roots.

Therefore:

- if f(z) has 1 root, then this root has multiplicity 3.

- If f(z) has 2 roots, then one has multiplicity 2 and the other has multiplicity 1.

- If f(z) has 3 roots, then all have multiplicity 1.

Furthermore, the number of real roots of f(z) can be 0, 1, 2, or 3.

(2) If f(z) has REAL COEFFICIENTS, then the number of real roots is 1, 2, or 3, and the non-real roots occur in complex conjugate pairs.

Therefore, if f(z) has real coefficients, then

- If f(z) has 1 root, it has to be real.

- if f(z) has two roots, both have to be real.

- If f(z) has three roots then either all are real, or one is real and two are non-real.