Question

If $2x=-1+\sqrt{3i}$, then the value of $(1-x^2+x{)}^6-(1-x+x^2{)}^6=$_______

• A. $32$
• B. $64$
• C. $-64$
• D. $0$

Answer 1

Answer: (D)

$0$

$2x=-1+\sqrt{3}i\Rightarrow x=\frac{-1+\sqrt{3}i}{2}=\omega$, cube root of unity

$\therefore (1-x^2+x{)}^6-(1-x+x^2{)}^6$

$=(1-{\omega }^2+\omega {)}^6-(1-\omega +{\omega }^2{)}^6$

$=(-2{\omega }^2{)}^6-(-\omega -\omega {)}^6$, since $1+\omega +{\omega }^2=0$

$=2^6{\omega }^{12}-2^6{\omega }^6$

$=2^6-2^6=0$, since ${\omega }^3=1$