To find the zeros of the function f(x) = 3x^3 - 4x^2 +8x+8, we need to solve for x when f(x) = 0.
One way to do this is to use synthetic division. We'll start by trying x = 1 as a possible zero:
1 | 3 -4 8 8
| 3 -1 7
| -----------
| 3 -1 7 15
Since the remainder is not zero, x = 1 is not a zero of the function. Let's try x = -1:
-1 | 3 -4 8 8
| -3 7 -15
| -----------
| 3 -7 15 -7
Since the remainder is zero, x = -1 is a zero of the function. We can now factor out (x + 1) from the polynomial using long division or synthetic division:
(x + 1)(3x^2 - 7x + 7)
The remaining quadratic factor does not have any real zeros, so the zeros of the function f(x) are:
x = -1 (with a multiplicity of 1)