If $\alpha, \beta, \gamma, \delta$ are the roots of the equation $x^{4}+x^{3}+x^{2}+x+1=0$, then $\alpha^{2021}+\beta^{2021}+\gamma^{2021}+\delta^{2021}$ is equal to :

<p>When, ${x^5} = 1$</p> <p>then ${x^5} - 1 = 0$</p> <p>$\Rightarrow (x - 1)({x^4} + {x^3} + {x^2} + x + 1) = 0$</p> <p>Given, ${x^4} + {x^3} + {x^2} + x + 1 = 0$ has roots $\alpha$, $\beta$, $\gamma$ and 8.</p> <p>$\therefore$ Roots of ${x^5} - 1 = 0$ are 1, $\alpha$, $\beta$, $\gamma$ and 8.</p> <p>We know, Sum of p^th power of n^th roots of unity = 0. (If p is not multiple of n) or n (If p is multiple of n)</p> <p>$\therefore$ Here, Sum of p^th power of n^th roots of unity</p> <p>$$ = {1^p} + {\alpha ^p} + {\beta ^p} + {\gamma ^p} + {8^p} = \left\{ {\matrix{ 0 & ; & {\mathrm{If\,p\,is\,not\,multiple\,of\,5}} \cr 5 & ; & {\mathrm{If\,p\,is\,multiple\,of\,5}} \cr } } \right.$$</p> <p>Here, $p = 2021$, which is not multiple of 5.</p> <p>$\therefore$ $${1^{2021}} + {\alpha ^{2021}} + {\beta ^{2021}} + {\gamma ^{2021}} + {8^{2021}} = 0$$</p> <p>$$ \Rightarrow {\alpha ^{2021}} + {\beta ^{2021}} + {\gamma ^{2021}} + {8^{2021}} = - 1$$</p> <p></p>