Find the value of $b$ for which $(2 x + 3)$ is factor of $2 x^{3} + 9 x^{2} - x - b .$

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Solution

$2 x + 3) ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ 2 x^{3} + 9 x^{2} - x - b (x^{2} + 3 x - 5$
$2 x^{3} + 3 x^{2} - --------- -$
$6 x^{2} - x - b$
$6 x^{2} + 9 x - -------- -$
$- 10 x - b$
$- 10 x - 15 - ---------- -$
$b + 15 - ----- -$

Since $2 x + 3$ in a factor of the given polynomial hence, remainder should be equal to $0.$
$∴$ $b + 15 = 0$
$b = - 15$

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