Question

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

Answer 1

$\text{When f(x) is divided by}(3x+2)\text{the remainder is}f\left(\frac{-2}{3}\right)\text{. Now,}$

$f\left(\frac{-2}{3}\right)=12{\left(\frac{-2}{3}\right)}^3-13{\left(\frac{-2}{3}\right)}^2–5\left(\frac{-2}{3}\right)+7$

$12\left(\frac{-8}{27}\right)-13\left(\frac{4}{9}\right)+\left(\frac{10}{3}\right)+7$
$\frac{-32}{9}-\frac{52}{9}+\frac{10}{3}+7=\frac{9}{9}=1$
Thus the remainder is 1.