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Square Root of a Complex Number

If ω is a c...

Question

If $ω$ is a complex cube root of unity and $x = ω^{2} - ω - 2,$ find the value of $x^{4} + 5 x^{3} + 9 x^{2} - x - 11.$

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Solution

Given,

$x^{4} + 5 x^{3} + 9 x^{2} - x - 11$

$x = w^{2} - w - 2$

$= {(w^{2} - w - 2)}^{4} + 5 {(w^{2} - w - 2)}^{3} + 9 {(w^{2} - w - 2)}^{2} - (w^{2} - w - 2) - 11$

$= (w^{8} - 4 w^{7} - 2 w^{6} + 20 w^{5} + w^{4} - 40 w^{3} - 8 w^{2} + 32 w + 16) + 5 (w^{6} - 3 w^{5} - 3 w^{4} + 11 w^{3} + 6 w^{2} - 12 w - 8) + 9 (w^{4} - 2 w^{3} - 3 w^{2} + 4 w + 4) - (w^{2} - w - 2) - 11$

$= w^{8} - 4 w^{7} + 3 w^{6} + 5 w^{5} - 5 w^{4} - 3 w^{3} - 6 w^{2} + 9 w + 3$

$= w^{2} - 4 w + 3 + 5 w^{2} - 5 w - 3 - 6 w^{9} w + 3$

$= 0 w^{2} + 0 w + 3$

$= 3$

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