Find $\alpha^4+\beta^4$ if $\alpha, \beta$ are the roots of equation $x^2+x+1=0$.

Question

Aicharya · Accepted Answer

The roots of the equation $x^2+x+1=0$ are $\alpha$ and $\beta$ and $\alpha^3=\beta^3=1$ and $\alpha+\beta+1=0$ $\{\because \alpha, \beta$ are non-real cube roots of unity $\}$ $\begin{aligned} \therefore \quad & \alpha^4+\beta^4=\alpha+\beta \ & =-1=-\alpha \beta \quad \quad\{\because \text { product of roots } \alpha \beta=1\} \end{aligned}$

Find \(\alpha^4+\beta^4\) if \(\alpha, \beta\) are the roots of equation \(x^2+x+1=0\).

Solution