The number of zeroes in a cubic polynomial

Definition of a cubic polynomial:
A cubic polynomial is a polynomial of degree three, which can be expressed in the form:
\[ax^3 + bx^2 + cx + d.\]

Fundamental theorem of algebra:
According to the fundamental theorem of algebra, a polynomial of degree 'n' has exactly 'n' roots/counting multiplicities in the complex number system.

Number of zeroes in a cubic polynomial:
1. A cubic polynomial has at most three zeroes.
2. The cubic polynomial may have three real roots or one real root and a pair of complex conjugate roots.
3. The number of zeroes can be determined by factoring the polynomial, setting it equal to zero, and solving for the roots.

Case 1: Three real roots
- If all three roots of the cubic polynomial are real, then the polynomial will have three distinct zeroes.

Case 2: One real root and a pair of complex conjugate roots
- If one root is real and the other two are complex conjugate, the polynomial will have one real zero and the complex conjugate pair.

Conclusion:
The number of zeroes in a cubic polynomial can be either three real roots or one real root and a pair of complex conjugate roots. The maximum number of zeroes in a cubic polynomial is three, corresponding to its degree.