• question_answer The remainder, when $12{{x}^{3}}-13{{x}^{2}}-5x+9$ is divided by $(3x+2),$ is A)  1                    B)  2 C)  3                    D)  4

Let $p(x)=12{{x}^{3}}-13{{x}^{2}}-5x+9$ and $q(x)=(3x+2)=3\,\left( x+\frac{2}{3} \right)$ $=3\,\left[ x-\left( \frac{-2}{3} \right) \right]$ When $P(x)$ is divided by $(3x+2)$ and the remainder is $P\,\left( -\frac{2}{3} \right)$ Now, $P\left( -\frac{2}{3} \right)=12\,{{\left( -\frac{2}{3} \right)}^{3}}-13\,{{\left( -\frac{2}{3} \right)}^{2}}-5\,\left( -\frac{2}{3} \right)+9$ $=12\times \left( \frac{-8}{27} \right)-13\times \frac{4}{9}+\frac{10}{3}+9$ $=-\frac{96}{27}-\frac{52}{9}+\frac{10}{3}+9$ $=\frac{-96-156+90+243}{27}=\frac{81}{27}=3$ $\therefore$ Required remainder = 3