No, every polynomial function of degree 3 with real coefficients does not necessarily have exactly three real zeros. In fact, a cubic polynomial can have one real zero and two complex zeros.
To understand this better, consider the general form of a cubic polynomial: f(x) = ax³ + bx² + cx + d, where a, b, c, and d are real coefficients and a is not zero.
According to the Fundamental Theorem of Algebra, a cubic polynomial will have exactly three roots (which can be real or complex) when counted with multiplicity. These roots can either be:
- Three distinct real roots,
- One real root and a pair of complex conjugate roots (which are non-real), or
- A repeated real root (with multiplicity) and one other distinct real root.
For example, the polynomial f(x) = x³ – 3x + 2 has three distinct real roots (at x = -1, 1, and 2). In contrast, the polynomial g(x) = x³ + 1 has only one real root (at x = -1) and two complex roots. Therefore, while cubic polynomials do have three roots in total, they do not need to have only real zeros.