Find the remainder when $f (x) = 12 x^{3} - 13 x^{2} - 5 x + 7$ is divided by $(3 x + 2) .$

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Solution

$f (x) = 12 x^{3} - 13 x^{2} - 5 x + 7$
When f(x) is divided by (3x+2) the remainder is $f (- \frac{2}{3})$ . Now,
$f (- \frac{2}{3}) = 12 {(- \frac{2}{3})}^{3} - 13 {(- \frac{2}{3})}^{2} - 5 (- \frac{2}{3}) + 7$
$= 12 (- \frac{8}{27}) - 13 (\frac{4}{9}) + \frac{10}{3} + 7$
$- \frac{32}{9} - \frac{52}{9} + \frac{10}{3} + 7 = \frac{9}{9} = 1$
$∴$ The remainder is 1.

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