By Descarte's rule of sign, the maximum number of negative real roots of equation $x^{5} - 6 x^{4} + 7 x^{3} + 8 x^{2} + 9 x + 10 = 0$

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Solution

Given equation is $x^{5} - 6 x^{4} + 7 x^{3} + 8 x^{2} + 9 x + 10 = 0$
$let f (x) = x^{5} - 6 x^{4} + 7 x^{3} + 8 x^{2} + 9 x + 10$
$\Rightarrow f (- x) = - x^{5} - 6 x^{4} - 7 x^{3} + 8 x^{2} - 9 x + 10$
Since the terms of $f (- x)$ are already in descending order.
Here number of sign changes in $f (- x)$ is 3.
Hence by Descarte's rule of signs, maximum number of negative real roots of $f (x)$ is 3.

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