"Let $\alpha$ be a root of the equation 1 + x2 + x4 = 0. Then, the value of $\alpha$1011 + $\alpha$2022 $-$ $\alpha$3033 is equal to :"

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Accepted Answer

"

Given, $\alpha$ is a root of the equation 1 + x² + x⁴ = 0

$\therefore$ $\alpha$ will satisfy the equation.

$\therefore$ 1 + $\alpha$² + $\alpha$⁴ = 0

${\alpha ^2} = {{ - 1 \pm \sqrt {1 - 4} } \over 2}$

$= {{ - 1 \pm \sqrt 3 i} \over 2}$

$\therefore$ ${\alpha ^2} = \omega \,ar\,{\omega ^2}$

Now,

${\alpha ^{1011}} + {\alpha ^{2022}} - {\alpha ^{3033}}$

$$ = \alpha \,.\,{({\alpha ^2})^{505}} + {({\alpha ^2})^{1011}} - \alpha \,.\,{({\alpha ^2})^{1516}}$$

$= \alpha {(\omega )^{505}} + {(\omega )^{1011}} - \alpha \,.\,{(\omega )^{1516}}$

$$ = \alpha \,.\,{({\omega ^3})^{168}}\,.\,\omega + {({\omega ^3})^{337}} - \alpha \,.\,{({\omega ^3})^{505}}\,.\,\omega $$

$= \alpha \,\omega + 1 - \alpha \,\omega$

$= 1$

"

Let $\alpha$ be a root of the equation 1 + x² + x⁴ = 0. Then, the value of $\alpha$¹⁰¹¹ + $\alpha$²⁰²² $-$ $\alpha$³⁰³³ is equal to :