question_answer

If \[\omega \]is a cube root of unity, then

A) \[{{x}^{3}}+1\]         

B) \[{{x}^{3}}+\omega \]

C) \[{{x}^{3}}+{{\omega }^{2}}\]        

D) \[{{x}^{3}}\]

Correct Answer: D

Solution :

[d]: Let   \[({{C}_{1}}\to {{C}_{1}}+{{C}_{2}}+{{C}_{3}})\] \[=x[1\{(x+{{\omega }^{2}})(x+\omega )-1\}+\omega \{1-(x+\omega )\}\] \[+{{\omega }^{2}}\{1-(x+{{\omega }^{2}})\}]\] \[=x({{x}^{2}}+\omega x+{{\omega }^{2}}x+{{\omega }^{3}}-1+\omega -\omega x-{{\omega }^{2}}+{{\omega }^{2}}\] \[-{{\omega }^{2}}x-{{\omega }^{4}})={{x}^{3}}\]              \[(\because {{\omega }^{3}}=1)\].