If 1, $w$ , $w^{2}$ are cube roots of unity , then find the value of ${(1 + 3 w + w^{2}) + (1 + 3 - w^{2})}^{4}$

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Solution

Given,

$1 + 3 w + w^{2} + {(1 + 3 - w^{2})}^{4}$

$= 1 + 3 w + w^{2} + {(- w^{2} + 4)}^{4}$

$= 1 + 3 w + w^{2} + w^{8} - 16 w^{6} + 96 w^{4} - 256 w^{2} + 256$

$= w^{8} - 16 w^{6} + 96 w^{4} - 255 w^{2} + 3 w + 257$

$= w^{2} - 16 + 96 w - 255 w^{2} + 3 w + 257$

$= 241 - 256 w^{2} + 99 w$

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