Using the remainder theorem, find the remainder when f(x) is divided by g(x) and verify the by actual division

Here, $f(x)=9 x^{3}-3 x^{2}+x-5$

g(x) = x − 2/3

from, the remainder theorem when f(x) is divided by g(x) = x - 2/3 the remainder will be equal to f(2/3)

substitute the value of x in f(x)

$f(2 / 3)=9(2 / 3)-3(2 / 3)^{2}+(2 / 3)-5$

= 9(8/27) − 3(4/9) + 2/3 − 5

= (8/3) − (4/3) + 2/3 − 5

$=\frac{8-4+2-15}{3}$

$=\frac{10-19}{3}$

= - 9/3

= - 3

Therefore, the remainder is - 3