Question

\[f(x) = 3 x^4 + 2 x^3 - \frac{x^2}{3} - \frac{x}{9} + \frac{2}{27}, g(x) = x + \frac{2}{3}\]

Answer 1

Let us denote the given polynomials as

`f(x) = 3x^4 + 2x^3 - x^2/3 - x/9 + 2/27,`

`g(x) = x+ 2/3`

`⇒ g(x) = x-(-2/3)`

We have to find the remainder when f(x) is divided by g(x).

By the remainder theorem, when f(x) is divided by g(x) the remainder is

`f(-2/3) =3 (-2/3)^4 + 2(-2/3)^3 - ((-2/3)^2) /3 - ((-2/3))/9 + 2/27`

                  ` = 3 xx 16 /81 - 2 xx 8/27 - 4/27 + 2/27 + 2/27`

                  ` = 16/27 - 16/27 - 4/27 + 2/27 + 2/27`

                   ` = 0`

Remainder by actual division

Remainder is 0