The given equation is $4x^3 + 9x^2 - x + 10 = 0$, and the problem asks which values are possible rational roots according to the Rational Root Theorem.

Rational Root Theorem

The theorem states that any rational root of a polynomial equation, given in the form $ax^n + bx^{n-1} + \cdots + k = 0$, must be a fraction $\frac{p}{q}$, where:

• $p$ is a factor of the constant term (here, 10).
• $q$ is a factor of the leading coefficient (here, 4).

Step 1: Identify the constant and leading coefficients.

• Constant term $k = 10$
• Leading coefficient $a = 4$

Step 2: List the factors.

• Factors of 10 (constant term): $\pm 1, \pm 2, \pm 5, \pm 10$
• Factors of 4 (leading coefficient): $\pm 1, \pm 2, \pm 4$

Step 3: List possible rational roots $\frac{p}{q}$.

The possible values of $\frac{p}{q}$, where $p$ is a factor of 10 and $q$ is a factor of 4, are: $\pm 1, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm 2, \pm 5, \pm \frac{5}{2}, \pm 10, \pm \frac{10}{2} = \pm 5$

Thus, the possible rational roots according to the Rational Root Theorem are: $\pm 1, \pm \frac{1}{2}, \pm \frac{5}{2}, \pm 2, \pm 5$

Step 4: Check the provided answers.

The possible answers are:

• $\pm \frac{5}{2}$ → Yes, this is valid.
• $\pm 2$ → Yes, this is valid.
• $\pm \frac{2}{3}$ → No, this is not valid.
• $\pm \frac{1}{2}$ → Yes, this is valid.

Final Answer:

The correct values are:

• $\pm \frac{5}{2}$
• $\pm 2$
• $\pm \frac{1}{2}$

Would you like more details or have any questions?

Here are 5 related questions to further your understanding:

1. How do you find the factors of a constant term and leading coefficient in the Rational Root Theorem?
2. Why is the Rational Root Theorem helpful for solving polynomial equations?
3. How can you check whether a rational root is an actual root of the polynomial?
4. What happens if none of the rational roots work? What methods could you use next?
5. How does synthetic division help when testing possible roots?

Tip: Always test the potential rational roots by substituting them into the equation or using synthetic division.