Key Concepts and Formulas

• Roots of Unity: The $n$-th roots of unity are the $n$ complex solutions to the equation $z^n = 1$.
• Sum of Powers of Roots of Unity: If $\omega_0, \omega_1, \dots, \omega_{n-1}$ are the $n$-th roots of unity, then $\sum_{k=0}^{n-1} \omega_k^p = \begin{cases} n & \text{if } p \equiv 0 \pmod{n} \\ 0 & \text{if } p \not\equiv 0 \pmod{n} \end{cases}$
• Factorization: $x^n - 1 = (x-1)(x^{n-1} + x^{n-2} + \dots + x + 1)$

Step-by-Step Solution

Step 1: Identifying the Roots

The given equation is $x^4 + x^3 + x^2 + x + 1 = 0$. We aim to recognize these roots as related to roots of unity.

Consider the equation $x^5 - 1 = 0$. We can factor this as: $x^5 - 1 = (x-1)(x^4 + x^3 + x^2 + x + 1)$

The roots of $x^5 - 1 = 0$ are the 5th roots of unity, given by $e^{i\frac{2\pi k}{5}}$ for $k = 0, 1, 2, 3, 4$. These are $1, e^{i\frac{2\pi}{5}}, e^{i\frac{4\pi}{5}}, e^{i\frac{6\pi}{5}}, e^{i\frac{8\pi}{5}}$.

The roots of $x^4 + x^3 + x^2 + x + 1 = 0$ are the 5th roots of unity excluding 1. This is because when $x=1$, $x^4 + x^3 + x^2 + x + 1 = 5 \neq 0$.

Therefore, $\alpha, \beta, \gamma, \delta$ are the non-unity 5th roots of unity.

Step 2: Applying the Sum of Powers Property

We want to find $\alpha^{2021} + \beta^{2021} + \gamma^{2021} + \delta^{2021}$.

Let's consider the sum of the 2021st powers of all 5th roots of unity (including 1): $1^{2021} + \alpha^{2021} + \beta^{2021} + \gamma^{2021} + \delta^{2021}$

Here, $n = 5$ (number of roots of unity) and $p = 2021$ (the power). We need to check if $p$ is a multiple of $n$.

Divide $2021$ by $5$: $2021 = 5 \cdot 404 + 1$ So, $2021 \equiv 1 \pmod{5}$. Since the remainder is 1, $2021$ is not a multiple of $5$.

Applying the sum of powers of roots of unity property: $1^{2021} + \alpha^{2021} + \beta^{2021} + \gamma^{2021} + \delta^{2021} = 0$

Step 3: Solving for the Required Sum

Since $1^{2021} = 1$, we have: $1 + \alpha^{2021} + \beta^{2021} + \gamma^{2021} + \delta^{2021} = 0$

Subtracting 1 from both sides gives: $\alpha^{2021} + \beta^{2021} + \gamma^{2021} + \delta^{2021} = -1$

Common Mistakes & Tips

• Recognizing roots of unity: Learn to quickly recognize the polynomial $x^{n-1} + x^{n-2} + \dots + x + 1$ as being intimately linked to the $n$-th roots of unity.
• Remember to exclude 1: Ensure you correctly identify which roots belong to the original equation. In this case, the root $x=1$ must be excluded.
• Modular Arithmetic: Use modular arithmetic to quickly determine if the power is a multiple of $n$.

Summary

By recognizing the roots of the given equation as the non-unity 5th roots of unity, we can leverage the property of the sum of powers of roots of unity. Since $2021 \equiv 1 \pmod{5}$, the sum of the 2021st powers of all 5th roots of unity is 0. This allows us to isolate the required sum, resulting in an answer of -1.

The final answer is \boxed{-1}, which corresponds to option (B).