The following function is given. $$ f(x)=x^{3}-3 x^{2}-25 x+75 $$ a. List all rational zeros that are possible according to the Rational Zero Theorem. $$ \pm 1, \pm 3, \pm 5, \pm 15, \pm 25, \pm 75 $$ (Use a comma to separate answers as needed.) b. Use synthetic division to test several possible rational zeros in order to identify one actual zero. One rational zero of the given function is (Simplify your answer.)

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To solve the problem, we will follow the steps outlined in parts (a) and (b).

### Part (a): List all rational zeros according to the Rational Zero Theorem.

The Rational Zero Theorem states that any rational solution (zero) of the polynomial equation $$ f(x) = 0 $$ is of the form $$ \frac{p}{q} $$, where $$ p $$ is a factor of the constant term (in this case, 75) and $$ q $$ is a factor of the leading coefficient (in this case, 1).

The factors of 75 are:
- $$ \pm 1, \pm 3, \pm 5, \pm 15, \pm 25, \pm 75 $$

Since the leading coefficient is 1, the possible rational zeros are simply the factors of 75.

Thus, the list of all possible rational zeros is:
$$
\pm 1, \pm 3, \pm 5, \pm 15, \pm 25, \pm 75
$$

### Part (b): Use synthetic division to test several possible rational zeros.

We will test some of the possible rational zeros using synthetic division. Let's start with $$ x = 3 $$:

1. Set up synthetic division with $$ 3 $$:

$$
\begin{array}{r|rrrr}
3 & 1 & -3 & -25 & 75 \
  &   & 3 & 0 & -75 \
\hline
  & 1 & 0 & -25 & 0 \
\end{array}
$$

The result of the synthetic division shows that the remainder is 0, which means $$ x = 3 $$ is indeed a zero of the polynomial $$ f(x) $$.

Thus, one rational zero of the given function is:
$$
3
$$
One rational zero of the given function is 3.

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