Final answer:
The fundamental theorem of algebra states that there will be 3 complex roots for the cubic function f(x) = 2x³ + 3x² - 4x - 12. To find these roots, the polynomial is set equal to zero, and one may use factoring, synthetic division, or other methods to determine the solutions.
Step-by-step explanation:
To use the fundamental theorem of algebra to state all the possible roots of the function f(x) = 2x³ + 3x² - 4x - 12, we recognize that the degree of this polynomial, which is 3, indicates that there will be 3 complex roots (counting multiplicity). These roots can be real or complex numbers. To find these roots, one might use techniques like factoring, synthetic division, or applying numerical methods if the roots are not easily factored.
Typically, these roots will be found by setting the polynomial equal to zero and solving for x to obtain the roots that satisfy the equation 2x³ + 3x² - 4x - 12 = 0. If the polynomial can be factored by grouping or using the rational root theorem, we can find the roots algebraically. Otherwise, numerical methods or graphing may be necessary to approximate the roots.