Final answer:
The cubic polynomial is factored by grouping, resulting in (5x² + 2)(x - 6). The real zero is found to be x = 6, while complex zeros can be obtained by applying the quadratic formula to the factor 5x² + 2.
Step-by-step explanation:
To factor the cubic polynomial f(x) = 5x³ - 30x² + 2x - 12 by grouping, we need to group terms that have common factors and then factor those common factors out. First, we group the terms as follows: (5x³ - 30x²) + (2x - 12). We can factor out a 5x² from the first group and a 2 from the second group, resulting in 5x²(x - 6) + 2(x - 6). Notice that both groups contain a common factor of (x - 6), which we can factor out as well:
f(x) = (5x² + 2)(x - 6).
Next, we see that 5x² + 2 does not factor further, so the factored form of the polynomial is as shown. To identify the zeros of f(x), we set each factor equal to zero and solve for x. From the first factor 5x² + 2, the equation is not readily solvable by simple methods but would require using the quadratic formula to find complex roots as the equation does not factor to real numbers. For the second factor x - 6, we get x = 6.