Question

If $x^2 + x + 1 = 0$, then $x^{2018} + x^{2019}$ equals which of the following:

Accepted Answer

C. Since $x^2 + x + 1 = 0$, $x$ is a cube root of unity $\omega$. Therefore $x^3 = 1$. $x^{2018} = x^{3 	imes 672 + 2} = x^2$. $x^{2019} = x^{3 	imes 673} = 1$. $x^{2018} + x^{2019} = x^2 + 1$. From $x^2 + x + 1 = 0 \Rightarrow x^2 + 1 = -x$.

Complex Roots of Unity

Explanation

Practise this under exam conditions