When $x^{3} + 3 x^{2} - k x + 4$ is divided by $(x - 2)$ , the remainder is $2 k$ then the value of $k$ is

6

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2

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Solution

The correct option is D $6$
When $x^{3} + 3 x^{2} - k x + 4$ is divided by $(x - 2)$ , the remainder is $2 k$
$∴ (x^{3} + 3 x^{2} - k x + 4) - 2 k$ is exactly divisible by $(x - 2)$
$\Rightarrow x = 2$ will satisfy the expression $[x^{3} + 3 x^{2} - k x + 4 - 2 k]$
$\Rightarrow$ ${(2)}^{3} + 3 {(2)}^{2} - k (2) + 4 - 2 k = 0$
$\Rightarrow$ $8 + 12 - 2 k + 4 - 2 k = 0$
$\Rightarrow 24 - 4 k = 0$
$\Rightarrow$ $k =$ $\frac{24}{4} = 6$

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