Key Concepts and Formulas

• Vieta's Formulas: For a quadratic equation $ax^2 + bx + c = 0$ with roots $\alpha$ and $\beta$, the sum of the roots is $\alpha + \beta = -\frac{b}{a}$ and the product of the roots is $\alpha \beta = \frac{c}{a}$.
• Relationship between Roots: Understanding how the roots are related to each other is crucial. In this case, one root is the cube of the other ($\beta = \alpha^3$).
• Fourth Root: The fourth root of a positive number has both positive and negative real solutions.

Step-by-Step Solution

Step 1: Define the Roots and Set Up the Problem

We are given the quadratic equation $81x^2 + kx + 256 = 0$. Let the roots be $\alpha$ and $\beta$, where $\beta = \alpha^3$. Our goal is to find a possible value of $k$.

Step 2: Apply Vieta's Formula for the Product of Roots

Using Vieta's formulas, the product of the roots is given by: $\alpha \beta = \frac{c}{a}$ Substituting $\beta = \alpha^3$, $a = 81$, and $c = 256$, we get: $\alpha (\alpha^3) = \frac{256}{81}$ $\alpha^4 = \frac{256}{81}$ Taking the fourth root of both sides: $\alpha = \pm \sqrt[4]{\frac{256}{81}} = \pm \frac{\sqrt[4]{256}}{\sqrt[4]{81}} = \pm \frac{4}{3}$ Thus, $\alpha$ can be either $\frac{4}{3}$ or $-\frac{4}{3}$.

Step 3: Apply Vieta's Formula for the Sum of Roots

Using Vieta's formulas, the sum of the roots is given by: $\alpha + \beta = -\frac{b}{a}$ Substituting $\beta = \alpha^3$, $a = 81$, and $b = k$, we get: $\alpha + \alpha^3 = -\frac{k}{81}$ $k = -81(\alpha + \alpha^3)$

Step 4: Calculate k for $\alpha = \frac{4}{3}$

If $\alpha = \frac{4}{3}$, then $\alpha^3 = \left(\frac{4}{3}\right)^3 = \frac{64}{27}$. Substituting into the equation for $k$: $k = -81\left(\frac{4}{3} + \frac{64}{27}\right) = -81\left(\frac{36}{27} + \frac{64}{27}\right) = -81\left(\frac{100}{27}\right)$ $k = -3(100) = -300$

Step 5: Calculate k for $\alpha = -\frac{4}{3}$

If $\alpha = -\frac{4}{3}$, then $\alpha^3 = \left(-\frac{4}{3}\right)^3 = -\frac{64}{27}$. Substituting into the equation for $k$: $k = -81\left(-\frac{4}{3} - \frac{64}{27}\right) = -81\left(-\frac{36}{27} - \frac{64}{27}\right) = -81\left(-\frac{100}{27}\right)$ $k = 3(100) = 300$

Step 6: Determine the Correct Option

We have found two possible values for $k$: $-300$ and $300$. The question asks for a value of $k$. Comparing these values with the given options, we see that option (B) is $-300$, which is one of our calculated values.

Common Mistakes & Tips

• Sign Errors: Pay close attention to signs, especially when dealing with negative roots and cubing them.
• Forgetting $\pm$: Remember to consider both positive and negative roots when taking the fourth root.
• Arithmetic: Double-check your arithmetic, especially when dealing with fractions.

Summary

We used Vieta's formulas to relate the roots and coefficients of the quadratic equation. By using the product of the roots, we found the possible values of $\alpha$. Then, using the sum of the roots, we found the possible values of $k$. The possible values for $k$ were $-300$ and $300$, and $-300$ is one of the answer choices.

The final answer is \boxed{-300}, which corresponds to option (B).