Which of the following should be added to $9 x^{3} + 6 x^{2} + x + 2$ so that the sum is divisible by (3x+1) ?

-4

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-3

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-2

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-1

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Solution

The correct option is D -2
Let k should be added to the given expression so that the sum is divisible by (3x+1).
Let $f (x) = 9 x^{3} + 6 x^{2} + x + 2 + k$
$f (\frac{- 1}{3}) = 0$

$9 (\frac{- 1}{3})^{3} + 6 (\frac{- 1}{3})^{2} - \frac{1}{3} + 2 + k = 0$

$2 + k = 0$
$k = - 2$

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