BY Remainder theorem , find the remainder when `p(x)` is divided by `g(x)`
(i) `p(x)` `=x^(3)-2x^(2)-4x-1, g(x)=x+1`
(ii) `p(x) =x^(3)-3x^(2)+4x+50, g(x) =x-3`

Explanation

(i) given `p(x)=x^(3)-2x^(2)-4x-1 and g(x) =x+1`
Here , zero of g(x) is -1
when we divide p(x) g(x) using remaining theorem , we get the remainder p-13).
`therefore p-(-1)=(-1)^(3)-2(-1)^(2)-4(-1)-1`
`=-1-2+4-1`
`=4-4=0`
Hence , remainder is 0.
(ii) GIven `-p(x)=x^(3)-3x+4x+50 and g(x) =x-3`
Here ,zero pf g(x) is 3.
when we divide p(x)by g(x) using remainder theoem , we get the remainderp(3).
`therefore p(3)=(3)^(3)-3(3)^(2)+4(3)+50`
`=27-27+12+50=62`
Hence , remainder is 62.
(iii) Given `p(x)=4x^(3)-12x^(2)+14x-3 and =2x-1`
here , zero of g(x) is `(1)/(2)`
when we divide p(x) by g(x) using reminder theorem, we get the remainder `p((1)/(2))`
` therefore p((1)/(2))=4((1)/(2))^(3)-12((1)/(2))^(2)+14((1)/(2))-3=4xx(1)/(8)-12xx(1)/(4)+14xx(1)/(2)-3`
`=(1)/(2)-3+7-3=(1)/(2)+1=(1+2)/(2)=(3)/(2)`
Hence , remainder is `(3)/(2)` .
(iv) Given `, p(x) =x^(3)-6x^(2)+2x-4and g(x) =1-(3)/(2)x.`
Here , zero of g(x) is `(2)/(3)`.
when we divide p(x) by g(x) using remainder theorem , we get the reminder `p((2)/(3))` .
`therefore =(8)/(27)-6xx(4)/(9) =2xx(2)/(3)-4=(8)/(27)-(24)/(9)+(4)/(3)-4`
`=(8-72+36-108)/(27) =(-136)/(27)`
hence remainder is `(-136)/(27)`