If a remainder of $4$ is obtained when $x^{3} + 2 x^{2} - x - k$ is divided by $x - 2$ , find the value of $k$ .

4

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6

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10

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12

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Solution

The correct option is B $10$
Let the quotient be $a x^{2} + b x + c$ , we get
$\Rightarrow x^{3} + 2 x^{2} - x - k = (x - 2) (a x^{2} + b x + c) + 4$
$= a x^{3} + (b - 2 a) x^{2} + (c - 2 b) x + 4 - 2 c$
By comparing, we get
$a = 1, b = 4, c = 7$
$\Rightarrow k = 14 - 4 = 10$

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