Final answer:
To find all zeros of the polynomial function, one must first recognize that x = -4 is a zero, divide the polynomial by (x + 4), and then apply the quadratic formula to find the remaining zeros of the resulting quadratic equation.
Step-by-step explanation:
The question deals with finding the zeros of a polynomial function algebraically. Given f(x) = 2x³ + 4x² - 6x - 40 and that f(-4) = 0, one of the zeros is at x = -4. Since -4 is a zero of the polynomial, (x + 4) is one of its factors. To find the other zeros, we can perform polynomial division by dividing the polynomial by (x + 4) to find the quotient, which will be a quadratic equation. We can then solve this quadratic equation to find the remaining zeros.
The solution process involves dividing the original polynomial f(x) by (x + 4), then applying the quadratic formula to the resulting quadratic equation ax² + bx + c = 0. The quadratic formula is x = (-b ± √(b² - 4ac)) / (2a).