Find the value of $b$ for which the polynomial $2 x^{3} - 9 x^{2} - x - b$ is divisible by $2 x + 3$

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Solution

$2 x + 3) 2 x^{3} - 9 x^{2} - x - b (x^{2} - 6 x + 17$
$2 x^{3} + 3 x^{2}$
______
$- 12 x^{2} - x - b$
$- 12 x^{2} - 18 x$
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$17 x - b$
$17 x + 51$
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$- b - 51$
_________
Remainder $= - b - 51 = 0$
$\Rightarrow b = 51$ for which the polynomial will be divisible by $2 x + 3$ .

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