If $x^{15} - x^{13} + x^{11} - x^{9} + x^{7} - x^{5} + x^{3} - x = 7$ then

is 15

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is less than 15

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greater than 15

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Solution

The correct option is C greater than 15
$x^{15} - x^{13} + x^{11} - x^{9} + x^{7} - x^{5} + x^{3} - x$
$= x (x^{8} + 1) (x^{4} + 1) (x^{2} - 1)$
$x^{16} - 1 = (x^{8} + 1) (x^{4} + 1) (x^{2} + 1) (x^{2} - 1)$
$= \frac{7}{x} (x^{2} + 1) = 7 [{(\sqrt{x} - \frac{1}{\sqrt{x}})}^{2} + 2] > 14$
$∴ x^{16} > 15$

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