If is a complex cube root of unity, show that : (2- ) (2- ^2 )=7 - Vidyadip

Question

If $\omega$ is a complex cube root of unity, show that : $(2-\omega)\left(2-\omega^2\right)=7$

✓

Answer

$\omega$ is the complex cube root of unity.
$
\therefore \omega^3=1 \text { and } 1+\omega+\omega^2=0
$
Also, $1+\omega^2=-\omega, 1+\omega=-\omega^2$ and $\omega+\omega^2=-1$
$
\begin{aligned}
& \text { L.H.S. }=(2-\omega)\left(2-\omega^2\right) \\
& =4-2 \omega^2-2 \omega+\omega^3 \\
& =4-2\left(\omega^2+\omega\right)+1 \\
& =4-2(-1)+1 \\
& =4+2+1 \\
& =7 \\
& =\text { R.H.S. }
\end{aligned}
$

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