Question

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

Answer 1

By Remainder theorem. if f(x) is divided by (x - a) then the remainder is f(a). 

Therefore,  if f(x) is divided by (ax + b) then the remainder is f(–b/a). 

If $f(x) = 12x^3 - 13x^2 - 5x - 7$ is divided by (3x + 2) then the remainder is $f (\frac{-2}{3} )$

 $(\frac{-2}{3} ) = 12( \frac{-2}{3})^3 - 13 (\frac{-2}{3})^2 - 5( \frac{-2}{3}) - 7$
                     
                          = $12( \frac{-8}{27}) - 13( \frac{4}{9}) + \frac{10}{3} - 7$
 
                          = $\frac{-32}{9} - \frac{52}{9} + \frac{10}{3} - \frac{7}{1}$

                          = $\frac{-32}{9} - \frac{52}{9} + \frac{30}{9} - \frac{63}{9}$

                          = $\frac{-117}{9}$
         
                          = –13

The remainder is –13