Number of distinct real roots of the equation $x^{4} + 4 x^{3} - 2 x^{2} - 12 x + k = 0$ is

4

x ϵ (- 7, 9)

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3

k = 7

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2

k < - 7

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no root if

k > 9

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Solution

The correct option is C $2$ If $k < - 7$
$\begin{matrix} x^{4} + 4 x^{3} - 2 x^{2} - 12 x + k = 0 x^{4} + 4 x^{3} - 2 x^{2} - 12 x = - k x (x^{3} + 4 x^{2} - 2 x - 12) = - k x (x + 2) (x^{2} + 2 x - 6) = - k N o w f (3) = - 3 (- 1) [9 - 6 - 6] f (- 1) = - 1 (1) [1 - 2 - 6] f (1) = 3 (- 3) = - 9 f o r - 9 < - k < 7 h a s 4 d i s t i n c t r e a l r o o t s - 7 < k < 9 f o r 2 d i s t i n c t r o o t - k > 7 k < - 7 F o r n o r o o t - k < - 9 k > 9 H e n c e, t h e o p t i o n C i s t h e c o r r e c t a n s w e r . \end{matrix}$

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