Answer:
[tex]{- \frac{238}{9}} [/tex]
Step-by-step explanation:
We want to find the remainder when the polynomial
[tex]p(x) = 3 {x}^{3} - 14 {x}^{2} - 14x + 3[/tex]
is divided by 3x+5.
According to the remainder theorem, a polynomial p(x) is divided by ax+b, then the remainder is given by
[tex]R(x) = p( - \frac{b}{a})[/tex]
We substitute x=-5/3 to get:
[tex]p( - \frac{5}{3} ) = 3 {(- \frac{5}{3})}^{3} - 14 {(- \frac{5}{3})}^{2} - 14 \times - \frac{5}{3}+ 3[/tex]
We now simplify to obtain:
[tex]p( - \frac{5}{3} ) = 3 {(- \frac{125}{27})} - 14 {( \frac{25}{9})} + \frac{70}{3}+ 3[/tex]
This simplifies to:
[tex]p( - \frac{5}{3} ) = - \frac{125}{9} - { \frac{350}{9}} + \frac{70}{3}+ 3[/tex]
[tex]p( - \frac{5}{3} ) = {- \frac{238}{9}} [/tex]
Therefore the remainder is -238/9
