To solve the questions based on the provided image:

1. Question (a): List all rational roots that are possible according to the Rational Zero Theorem.

   The Rational Zero Theorem states that any rational root of a polynomial equation $f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_0$ will be in the form $\pm \frac{p}{q}$, where $p$ is a factor of the constant term $a_0$ and $q$ is a factor of the leading coefficient $a_n$.

   Here, the polynomial is: $8x^3 + 50x^2 - 41x + 7 = 0$

   • Constant term $a_0 = 7$ with factors $\pm 1, \pm 7$.
   • Leading coefficient $a_n = 8$ with factors $\pm 1, \pm 2, \pm 4, \pm 8$.

   Possible rational roots are all combinations of $\frac{p}{q}$ with $p$ from $\pm 1, \pm 7$ and $q$ from $\pm 1, \pm 2, \pm 4, \pm 8$. This results in: $\pm 1, \pm 7, \pm \frac{1}{2}, \pm \frac{7}{2}, \pm \frac{1}{4}, \pm \frac{7}{4}, \pm \frac{1}{8}, \pm \frac{7}{8}$ Therefore, the correct answer is Option D.

2. Question (b): Use synthetic division to test several possible rational roots in order to identify one actual root.

   Synthetic division can be used to test each of the possible rational roots listed. A quick approach involves trying $x = 1$, $x = -1$, or simple fractions first, as they are often easier to calculate by hand.

   Testing these roots using synthetic division, we find that one rational root is $x = \frac{1}{2}$, as it simplifies the polynomial to zero. This root satisfies the equation, as requested.

Would you like a detailed breakdown of the synthetic division process for this root, or do you have other questions?

Here are 5 related questions to explore further:

1. How does synthetic division help in finding other roots after identifying one root?
2. How would you use the Rational Zero Theorem with different polynomials, such as fourth-degree polynomials?
3. What are the steps to factorize a polynomial after finding one of its roots?
4. How does the Rational Zero Theorem help reduce the number of potential roots to test?
5. What alternative methods exist for finding roots of polynomials, aside from synthetic division?

Tip: Always start with simpler factors (like 1 or -1) when using the Rational Zero Theorem, as they often reveal roots faster and simplify calculations.