In each of the following, using the remainder theorem, find the remain

1.	In each of the following, using the remainder theorem, find the remainder when f(x) is divided by g(x):f(x) = 9x3 - 3x2 + x - 5, g(x) = x= \(-\frac{2}{3}\)
Answer» We have, f(x) = 9x³ - 3x² + x - 5 and g(x) = x = \(-\frac{2}{3}\) Therefore, by remainder theorem when f (x) is divided by g (x) = x - \(\frac{2}{3},\) the remainder is equal to f \((\frac{2}{3})\) Now, f(x) = 9x³ - 3x² + x - 5 f \((\frac{2}{3})=9(\frac{2}{3})^{3}-3(\frac{2}{3})^{2}+\frac{2}{3}-5\) \(=\frac{8}{3}-\frac{4}{3}+\frac{2}{3}-5\) = 2 – 5 = - 3 Hence, the required remainder is - 3.

In each of the following, using the remainder theorem, find the remainder when f(x) is divided by g(x):f(x) = 9x3 - 3x2 + x - 5, g(x) = x= \(-\frac{2}{3}\)

Answer»

We have,

f(x) = 9x³ - 3x² + x - 5 and g(x) = x = \(-\frac{2}{3}\)

Therefore, by remainder theorem when f (x) is divided by g (x) = x - \(\frac{2}{3},\) the remainder is equal to f \((\frac{2}{3})\)

Now, f(x) = 9x³ - 3x² + x - 5

f \((\frac{2}{3})=9(\frac{2}{3})^{3}-3(\frac{2}{3})^{2}+\frac{2}{3}-5\)

\(=\frac{8}{3}-\frac{4}{3}+\frac{2}{3}-5\)

= 2 – 5 = - 3

Hence, the required remainder is - 3.