Using the remainder theorem, find out whether g(x) is factor of f(x) or not? `f(x)=2x^3-9x^2+x+12 ,\ g(x)=3-2x`

Explanation

`f(x)=2x^3-9x^2+x+12`
`g(x)=(3-2x)`
`(3-2x)=0`
`=>2x=3`
`=>x=3/2`.
`f(3/2)=0`
By remainder theorem, `(3 - 2x)` is the factor of `f(x)` if `f(3/2)=0`
Now,
`f(3/2)=2(3/2)^3-9(3/2)^2+(3/2)+12`
`= 27^4-81^4+3^2+12`
`=54^4+3^2+12`
`=-27^2+3^2+12`
`=-12+12`
`=0`
`therefore (3-2x)` is the factor of polynomial `f(x)`.