Final answer:
The zeros of f(x) = x^3 - 4x^2 - 5x + 20 are 4 and (-1 +/- sqrt(21)) / 2.
Step-by-step explanation:
The given function is f(x) = x^3 - 4x^2 - 5x + 20. To find the zeros of the function, we need to solve the equation f(x) = 0. One way to find the zeros is by using synthetic division or polynomial long division. However, in this case, we can use factoring by grouping to simplify the equation:
f(x) = (x^3 - 5x) + (-4x^2 + 20)
f(x) = x(x^2 - 5) + (-4)(x^2 - 5)
f(x) = (x - 4)(x^2 + x - 5)
Setting each factor equal to zero, we have:
x - 4 = 0 ---> x = 4
x^2 + x - 5 = 0
Using the quadratic formula, we can find the solutions of the quadratic equation:
x = (-b +/- sqrt(b^2 - 4ac)) / (2a)
Plugging in the values a = 1, b = 1, and c = -5, we get:
x = (-1 +/- sqrt(1^2 - 4(1)(-5))) / (2(1))
x = (-1 +/- sqrt(1 + 20)) / 2
x = (-1 +/- sqrt(21)) / 2
So, the zeros of f(x) = x^3 - 4x^2 - 5x + 20 are x = 4 and x = (-1 +/- sqrt(21)) / 2.