If $a$ and $b$ are real numbers such that $(2+\alpha)^4=a+b \alpha$, where $\alpha=\left(\frac{1+i \

Question

If $a$ and $b$ are real numbers such that $(2+\alpha)^4=a+b \alpha$, where $\alpha=\left(\frac{1+i \sqrt{3}}{2}\right)$, then $a+b=$

Professor PC Thomas Classes · Answer

$\alpha = -\omega^2$. Expand $(2-\omega^2)^4$ using $\omega^3=1$ and $1+\omega+\omega^2=0$ to find integer coefficients $a,b$. Sum is 9.