Final answer:
To find the zeros of the function f(x) = x^3 + 5x^2 - 22x + 16 algebraically, substitute -8 for x in the equation and solve. The solutions to the quadratic equation are x = -4 and x = 5.
Step-by-step explanation:
To find the zeros of the function f(x) = x^3 + 5x^2 - 22x + 16 algebraically, we need to solve the equation f(x) = 0.
Given that f(-8) = 0, we substitute -8 for x in the equation and solve: (-8)^3 + 5(-8)^2 - 22(-8) + 16 = 0.
Simplifying the equation, we get a quadratic equation: x^2 + 23x - 20 = 0.
We can solve this quadratic equation using factoring, completing the square, or using the quadratic formula.
The solutions to the quadratic equation are x = -4 and x = 5.