Key Concepts and Formulas

• Cube Roots of Unity: $\omega = \frac{-1+i\sqrt{3}}{2}$ is a non-real cube root of unity.
• Properties of $\omega$: $\omega^3 = 1$, $1 + \omega + \omega^2 = 0$.
• Binomial Theorem: $(x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k}y^k$

Step-by-Step Solution

• Step 1: Recognize and Substitute $\alpha$ as $\omega$ Since $\alpha = \frac{-1+i\sqrt{3}}{2}$, we recognize that $\alpha$ is a cube root of unity, commonly denoted as $\omega$. This allows us to use the properties of $\omega$ to simplify the expression. Substituting $\alpha = \omega$ into the given equation $(2 + \alpha)^4 = a + b\alpha$, we get: $(2 + \omega)^4 = a + b\omega$

• Step 2: Expand $(2+\omega)^4$ using the Binomial Theorem We use the binomial theorem to expand $(2+\omega)^4$: $(2+\omega)^4 = \binom{4}{0}2^4\omega^0 + \binom{4}{1}2^3\omega^1 + \binom{4}{2}2^2\omega^2 + \binom{4}{3}2^1\omega^3 + \binom{4}{4}2^0\omega^4$ $(2+\omega)^4 = (1)(16)(1) + (4)(8)(\omega) + (6)(4)(\omega^2) + (4)(2)(\omega^3) + (1)(1)(\omega^4)$ $(2+\omega)^4 = 16 + 32\omega + 24\omega^2 + 8\omega^3 + \omega^4$ The binomial theorem ensures we correctly account for all terms in the expansion.

• Step 3: Simplify Powers of $\omega$ using $\omega^3 = 1$ We use the property $\omega^3 = 1$ to simplify the powers of $\omega$. Since $\omega^3 = 1$, then $\omega^4 = \omega^3 \cdot \omega = 1 \cdot \omega = \omega$. Substituting these simplifications into the equation: $(2+\omega)^4 = 16 + 32\omega + 24\omega^2 + 8(1) + \omega$ $(2+\omega)^4 = 16 + 32\omega + 24\omega^2 + 8 + \omega$ This simplification reduces the higher powers of $\omega$ to either $\omega$ or $\omega^2$.

• Step 4: Combine Like Terms Combine the constant terms, $\omega$ terms, and $\omega^2$ terms: $(2+\omega)^4 = (16+8) + (32\omega+\omega) + 24\omega^2$ $(2+\omega)^4 = 24 + 33\omega + 24\omega^2$ This makes it easier to apply the property $1+\omega+\omega^2 = 0$.

• Step 5: Apply the Property $1+\omega+\omega^2=0$ to eliminate $\omega^2$ From the property $1+\omega+\omega^2=0$, we can write $\omega^2 = -1-\omega$. Substituting this into the expression: $(2+\omega)^4 = 24 + 33\omega + 24(-1-\omega)$ $(2+\omega)^4 = 24 + 33\omega - 24 - 24\omega$ This step is crucial for expressing the left-hand side in the form $a + b\omega$, which matches the right-hand side of the equation.

• Step 6: Simplify and Equate Coefficients Combine the remaining terms: $(2+\omega)^4 = (24-24) + (33\omega - 24\omega)$ $(2+\omega)^4 = 0 + 9\omega$ $(2+\omega)^4 = 9\omega$ Equating this with the given form $a+b\omega$: $9\omega = a+b\omega$ Comparing the coefficients of $1$ and $\omega$: $a = 0$ $b = 9$ We can equate coefficients because $1$ and $\omega$ are linearly independent.

• Step 7: Calculate $a+b$ Finally, we find the sum $a+b$: $a+b = 0 + 9$ $a+b = 9$

Common Mistakes & Tips

• Memorize and Correctly Apply Cube Root of Unity Properties: The properties $\omega^3=1$ and $1+\omega+\omega^2=0$ are essential. Make sure to apply them correctly.
• Pay attention to signs during substitutions. A common mistake is sign errors when substituting $\omega^2 = -1 - \omega$.
• Systematic Simplification: Simplify powers of $\omega$, combine like terms, and then eliminate $\omega^2$ terms using $1+\omega+\omega^2=0$.

Summary

The problem tests understanding of cube roots of unity. By recognizing $\alpha$ as $\omega$, expanding using the binomial theorem, and simplifying using the properties $\omega^3=1$ and $1+\omega+\omega^2=0$, the expression is reduced to $9\omega$. Comparing this to $a+b\omega$, we find $a=0$ and $b=9$, so $a+b=9$.

Final Answer The final answer is \boxed{9}, which corresponds to option (B).