To use the Rational Zero Theorem to find all the zeros of the polynomial function f(x) = x^4 + 2x^3 - 7x^2 - 8x + 12, we need to find the possible rational zeros.
The Rational Zero Theorem states that if a rational number p/q is a zero of the polynomial with integer coefficients, then p must be a factor of the constant term (in this case, 12), and q must be a factor of the leading coefficient (in this case, 1).
So, the possible rational zeros are determined by the factors of 12. They are: ±1, ±2, ±3, ±4, ±6, ±12.
Now, we can test each of these values by using synthetic division or long division to check if they are zeros of the polynomial. We are looking for values that give a remainder of 0 when the polynomial is divided by (x - value).
Using synthetic division, let's test if x = 1 is a zero:
1 | 1 2 -7 -8 12
-- 3 -4 -11 -19
-----------------
1 5 -11 -19 -7
Since the remainder is not 0, x = 1 is not a zero of the polynomial.
Continuing this process, we can test the other possible rational zeros (±2, ±3, ±4, ±6, ±12) until we find the zeros of the polynomial.
After testing all the possible rational zeros, we find that the polynomial function f(x) = x^4 + 2x^3 - 7x^2 - 8x + 12 has no rational zeros over the real numbers.