The number of distinct real roots of the equation $x^{5} (x^{3} - x^{2} - x + 1) + x (3 x^{3} - 4 x^{2} - 2 x + 4) - 1 = 0$ is

3.00

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3.0

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Solution

$x^{8} - x^{7} - x^{6} + x^{5} + 3 x^{4} - 4 x^{3} - 2 x^{2} + 4 x - 1 = 0$

$\Rightarrow x^{7} (x - 1) - x^{5} (x - 1) + 3 x^{3} (x - 1) - x (x^{2} - 1) + 2 x (1 - x) + (x - 1) = 0$

$\Rightarrow (x - 1) (x^{7} - x^{5} + 3 x^{3} - x (x + 1) - 2 x + 1) = 0$

$\Rightarrow (x - 1) (x^{7} - x^{5} + 3 x^{3} - x^{2} - 3 x + 1) = 0$

$\Rightarrow (x - 1) (x^{5} (x^{2} - 1) + 3 x (x^{2} - 1) - 1 (x^{2} - 1)) = 0$

$\Rightarrow (x - 1) (x^{2} - 1) (x^{5} + 3 x - 1) = 0$

$∴ x = \pm 1$ are roots of the above equation and $x^{5} + 3 x - 1$ is a monotonic term hence vanishs at exactly one value of x other then $1$ or $- 1$ .

$∴ 3$ real roots.

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