Question
If $$\omega$$ is a complex cube root of unity and $$x\, = {\omega ^2} - \omega - 2,$$ then what is the value of $$x^2 + 4x + 7\, ?$$
A.
$$- 2$$
B.
$$- 1$$
C.
$$0$$
D.
$$1$$
Answer :
$$0$$
Solution :
$$\eqalign{
& {\text{Given,}}\,x\, = {\omega ^2} - \omega - 2 \cr
& \Rightarrow \,x + 2 = {\omega ^2} - \omega \cr} $$
On squaring both sides, we get
$$\eqalign{
& {\left( {x + 2} \right)^2} = {\left( {{\omega ^2} - \omega } \right)^2} \cr
& \Rightarrow \,{x^2} + 4x + 4 = {\omega ^4} + {\omega ^2} - 2{\omega ^3} \cr} $$
Add 3 on both side
$$\eqalign{
& \Rightarrow \,{x^2} + 4x + 4 + 3 = \omega + {\omega ^2} - 2 + 3\,\,\left( {\because \,{\omega ^3} = 1} \right) \cr
& \Rightarrow \,{x^2} + 4x + 7 = 1 + \omega + {\omega ^2} \cr
& \Rightarrow \,{x^2} + 4x + 7 = 0\,\,\left( {\because \,1 + \omega + {\omega ^2} = 0} \right) \cr} $$