The equation whose roots are negative of those of $x^{n} + x^{n - 2} + x^{n - 5} + m = 0$ , where $n$ is even and $n \geq 6$ , is:

x^{n} - x^{n - 2} + x^{n - 5} + m = 0

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x^{n} + x^{n - 2} - x^{n - 5} + m = 0

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x^{n} + x^{n - 2} + x^{n - 5} + m = 0

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x^{n} - x^{n - 2} - x^{n - 5} + m = 0

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Solution

The correct option is D $x^{n} + x^{n - 2} - x^{n - 5} + m = 0$
Replacing $x \to - x$ in $x^{n} + x^{n - 2} + x^{n - 5} + m = 0$
${(- x)}^{n} + {(- x)}^{n - 2} + {(- x)}^{n - 5} + m = 0$
As $n$ is even $\Rightarrow$ $n - 2$ is even and $n - 5$ is odd
$\Rightarrow x^{n} + x^{n - 2} - x^{n - 5} + m = 0$

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