Final answer:
The polynomial x³ - 6x² + 4x - 24 = 0 has exactly 3 solutions according to the fundamental theorem of algebra since it is a third-degree polynomial. The correct option is C.
Step-by-step explanation:
The polynomial given in the question seems to have a typo, but we'll address the polynomial x³ - 6x² + 4x - 24 = 0. To determine the number of solutions this polynomial has, we can use the fundamental theorem of algebra, which states that a polynomial of degree n will have exactly n complex roots (solutions), counting multiplicities. Since the given polynomial is of degree 3 (x³ being the highest power), it has 3 solutions. These solutions could be real or complex numbers.
To solve for the exact solutions, we could factor the polynomial if possible, or apply methods like synthetic division or the rational root theorem. Often, graphical or numerical methods such as Newton's method might be used to approximate the roots.
However, it's important to note that even without solving for the exact values of the roots, we can confidently say that there are three solutions based on the polynomial's degree.