Question

If α and β are the roots of x² + x + 1 = 0, then what is \(\mathop \sum \limits_{j = 0}^3 \left( {{\alpha ^j} + {\beta ^j}} \right)\) equal to?
1. 8
2. 6
3. 4
4. 2

Answer 1

Correct Answer - Option 4 : 2

Calculation:

Given: x² + x + 1 = 0, here a = 1, b = 1 and c = 1.

\(\Rightarrow x = \frac{{ - 1 \pm \sqrt {1 - 4} }}{2} = \frac{{ - 1 \pm \sqrt 3 i}}{2}\)

⇒ The roots of the given equation are ω and ω².

\(\Rightarrow \mathop \sum \limits_{j = 0}^3 \left( {{\alpha ^j} + {\beta ^j}} \right) = 2 + \omega + {\omega ^2} + {\omega ^2} + {\left( {{\omega ^2}} \right)^2} + {\omega ^3} + {\left( {{\omega ^2}} \right)^3} = 2\)