The order of multiple roots of $2$ of the equation $x^{4} - 5 x^{3} + 6 x^{2} + 4 x - 8 = 0$ is

1

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2

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3

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4

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Solution

The correct option is B $3$
$x^{4} - 5 x^{3} + 6 x^{2} + 4 x - 8 = 0$
Let $f (x) = x^{4} - 5 x^{3} + 6 x^{2} + 4 x - 8 = 0$
Since, 2 is a multiple root of f(x)=0
$f (2) = 0$
$f^{'} (x) = 4 x^{3} - 15 x^{2} + 12 x + 4$
$f^{'} (2) = 0$
$f^{''} (x) = 12 x^{2} - 30 x + 12$
$f^{''} (2) = 0$
$f^{'''} (x) = 24 x - 30$
$f^{'''} (2) \neq 0$
Hence the order of multiple root 2 is 3.

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