Question

The remainder when $f\left(x\right)=2x^4+2x^3-\frac{x^2}{3}-\frac{x}{9}+\frac{2}{27}$ is divided by $g\left(x\right)=x+\frac{2}{3}$ is:

• A. $\frac{32}{81}$
• B. $1$
• C. $0$
• D. $\frac{-16}{81}$

Answer 1

The correct option is D $\frac{-16}{81}$

The polynomial remainder theorem says that when $f\left(x\right)$ is divided by $(x-a)$, the remainder is given by $f\left(a\right)$.

Now, $f\left(x\right)=2x^4+2x^3-\frac{x^2}{3}-\frac{x}{9}+\frac{2}{27}$ is divided by $g\left(x\right)=x+\frac{2}{3}$

Here, $a=-\frac{2}{3}$

$\therefore Remainder=f(-\frac{2}{3})=2{(-\frac{2}{3})}^4+2{(-\frac{2}{3})}^3-({(-\frac{2}{3})}^2\times \frac{1}{3})-((-\frac{2}{3})\times \frac{1}{9})+\frac{2}{27}$

$=-\frac{16}{81}$