The Rational Zero Theorem states that any rational zero of a polynomial function is of the form:

$\frac{p}{q}$

where:

• $p$ is a factor of the constant term (the last term of the polynomial).
• $q$ is a factor of the leading coefficient (the coefficient of the highest-degree term).

For the polynomial $P(x) = 2x^3 + x^2 - 49x + 8$:

1. The constant term is $8$, so the factors of $8$ are: $\pm 1, \pm 2, \pm 4, \pm 8$

2. The leading coefficient is $2$, so the factors of $2$ are: $\pm 1, \pm 2$

Now, form all possible rational zeros by taking each factor of the constant term and dividing it by each factor of the leading coefficient. The possible rational zeros are:

$\pm \frac{1}{1}, \pm \frac{2}{1}, \pm \frac{4}{1}, \pm \frac{8}{1}, \pm \frac{1}{2}, \pm \frac{2}{2}, \pm \frac{4}{2}, \pm \frac{8}{2}$

Simplifying these fractions gives:

$\pm 1, \pm 2, \pm 4, \pm 8, \pm \frac{1}{2}$

Thus, the possible rational zeros are:

$\pm 1, \pm 2, \pm 4, \pm 8, \pm \frac{1}{2}$

These are the possible rational zeros for the polynomial $P(x) = 2x^3 + x^2 - 49x + 8$.

Would you like more details or have any questions?

Here are some related questions for further practice:

1. Verify which of these rational zeros are actual solutions.
2. Use synthetic division to test one of the possible zeros.
3. What is the factorized form of the polynomial?
4. How does the Rational Zero Theorem help in solving polynomial equations?
5. What happens if a polynomial has no rational zeros?

Tip: Always check the factors of both the constant and leading terms carefully, as missing a factor could result in missing potential zeros!