Let's work through the solution step-by-step.
### Part (a)
The Rational Zero Theorem states that any rational zero of a polynomial function [tex]\( p(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \)[/tex] will be a fraction [tex]\( \frac{p}{q} \)[/tex], where [tex]\( p \)[/tex] is a factor of the constant term [tex]\( a_0 \)[/tex] and [tex]\( q \)[/tex] is a factor of the leading coefficient [tex]\( a_n \)[/tex].
For the polynomial [tex]\( f(x) = x^3 - 3x^2 - 4x + 12 \)[/tex]:
- The constant term [tex]\( a_0 \)[/tex] is 12.
- The leading coefficient [tex]\( a_n \)[/tex] is 1.
Factors of the constant term 12 are: [tex]\( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12 \)[/tex].
Factors of the leading coefficient 1 are: [tex]\( \pm 1 \)[/tex].
Thus, all possible rational zeros are given by the fractions of these factors:
[tex]\[ \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12 \][/tex]
So, the possible rational zeros are:
[tex]\[ -1, 1, -2, 2, -3, 3, -4, 4, -6, 6, -12, 12 \][/tex]
### Part (b)
To identify one actual zero, we test the possible rational zeros through synthetic division.
After testing several values, we find that [tex]\(-2\)[/tex] is a zero of the given function. This can be verified by substituting [tex]\(-2\)[/tex] into the polynomial equation and checking if [tex]\( f(-2) = 0 \)[/tex].
So, one rational zero of the given function is:
[tex]\[ -2 \][/tex]
### Part (c)
We use the zero from part (b) to factor the polynomial further and find all the zeros.
We already know that [tex]\(-2\)[/tex] is a zero. By factoring [tex]\( x + 2 \)[/tex] from [tex]\( f(x) \)[/tex] and simplifying, we find the remaining polynomial. This polynomial can be further factored or solved to obtain all zeros.
The remaining factorable polynomial derived from synthetic division (or by other algebraic methods) will yield the other zeros.
Combining all found zeros gives us:
[tex]\[ -2, -4, 0, 1 \][/tex]
Thus, the zeros of the function [tex]\( f(x) = x^3 - 3x^2 - 4x + 12 \)[/tex] are:
[tex]\[ -2, -4, 0, 1 \][/tex]
So, the full solution to the problem is as follows:
### Part (a)
[tex]\[ -1, 1, -2, 2, -3, 3, -4, 4, -6, 6, -12, 12 \][/tex]
### Part (b)
One rational zero of the given function is:
[tex]\[ -2 \][/tex]
### Part (c)
The zeros of the function [tex]\( f(x) = x^3 - 3x^2 - 4x + 12 \)[/tex] are:
[tex]\[ -2, -4, 0, 1 \][/tex]
