Final answer:
To determine all the zeros of the given function f(x) = x³ - 3x² - 5x + 39, we can use the fact that f(-3) = 0. This means that -3 is one of the zeros of the function. To find the other zeros, we can divide the function by (x + 3) using long division or synthetic division. The quotient will be a quadratic equation, which can be solved using factoring or the quadratic formula.
Step-by-step explanation:
To determine all the zeros of the given function f(x) = x³ - 3x² - 5x + 39, we can use the fact that f(-3) = 0. This means that -3 is one of the zeros of the function. To find the other zeros, we can divide the function by (x + 3) using long division or synthetic division. The quotient will be a quadratic equation, which can be solved using factoring or the quadratic formula.
Using long division, we find that the quotient is x² - 6x + 13. To find the zeros of this quadratic equation, we can use the quadratic formula: x = (-b ± √(b² - 4ac)) / (2a). Plugging in the values of a = 1, b = -6, and c = 13 into the formula, we get two complex conjugate zeros: x ≈ 3 + 2i and x ≈ 3 - 2i. Therefore, the zeros of the function f(x) are -3, 3 + 2i, 3 - 2i.