Given $\alpha$ , $\beta$ respectively the fifth and the fourth non-real roots of unity then the value of $\left(1+\alpha \right)\left(1+\beta \right)\left(1+\alpha ^2\right)\left(1+\beta ^2\right)\left(1+\alpha ^3\right)\left(1+\beta ^3\right)\left(1+\alpha ^4\right)$ will be ?
After seeing this question on first sight i though it would be an easy one as $x^4 -1 = 0$ and i will then apply $(x-1)(x-\alpha)…(x-\alpha ^{n-1}))$ but there is + sign between expression so i thought multiplying and dividing $(1-\alpha)$ and $(1-\beta)$ would help but the $(1+\alpha^{3})$ term is making trouble for me .
Options are (A) $0$ (B) $1$ (C) $(1+\alpha+\alpha^{2})(1-\beta^{2})$ (D) $(1+\alpha)(1+\beta+\beta^{2})$
Best Answer
Looking at the $\beta$ terms alone, multiplying the first two brackets gives $$(1+\beta)(1+\beta^2)=1+\beta+\beta^2+\beta^3=0$$ so the answer is A