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