The given equation is:

$x^3 - 14x - 8 = 0$

(a) List all rational roots possible according to the Rational Zero Theorem:

The Rational Zero Theorem helps to find possible rational roots by considering the factors of the constant term (here, $-8$) and the leading coefficient (here, $1$).

The possible rational roots are the factors of the constant term divided by the factors of the leading coefficient:

• Factors of $-8$ are $\pm 1, \pm 2, \pm 4, \pm 8$.
• Factors of $1$ are just $\pm 1$.

Thus, the possible rational roots are: $\pm 1, \pm 2, \pm 4, \pm 8$

You correctly selected option B: $\pm 1, \pm 2, \pm 4, \pm 8$.

(b) Use synthetic division to find one rational root:

To identify an actual root, we need to perform synthetic division. We will test possible roots from the list in part (a). Let's test $x = 2$ using synthetic division.

1. Coefficients of the polynomial: $[1, 0, -14, -8]$ (note the missing $x^2$ term is represented by 0).
2. Set up synthetic division with $2$:

2 |  1    0   -14   -8
    |       2     4   -20
    ---------------------
       1    2   -10  -28

The remainder is $-28$, meaning $2$ is not a root. Let's test $x = -2$:

-2 |  1    0   -14   -8
     |      -2    4   20
     ---------------------
        1   -2   -10   12

The remainder is $12$, meaning $-2$ is also not a root. Continuing this way for other values like $\pm 1$ should eventually find a rational root.

You can now try testing $x = 1$ or similar values next.

Would you like to explore synthetic division steps for other potential roots? Here are some relevant follow-up questions:

1. How does synthetic division work for polynomials of higher degrees?
2. What happens if none of the possible rational roots from the theorem work?
3. How can the Rational Zero Theorem assist with factoring polynomials?
4. Can synthetic division be applied to non-rational roots?
5. How does synthetic division relate to polynomial division methods?

Tip: When using synthetic division, if a root works, the remainder will be 0.