Final answer:
The cubic function with zeros √6, -√6, and -3 is f(x) = x³ + 3x² - 6x - 18.
Step-by-step explanation:
The student is looking to find a cubic function with the given zeros: √6, -√6, and -3. To find the cubic function, we construct the factors associated with each zero.
The factors for the zeros √6 and -√6 are (x - √6)(x + √6) which simplifies to , and the factor for the zero -3 is (x + 3).
To find the cubic function, we multiply these factors together: (x² - 6)(x + 3). Expanding this, we get:
x(x² - 6) + 3(x² - 6) = x³ - 6x + 3x² - 18 = x³ + 3x² - 6x - 18.
Therefore, the correct cubic function is f(x) = x³ + 3x² - 6x - 18.