Show that
(i) `x+3 ` is factor of `69+11x-x^(2)+x^(3)`
(ii) `2x-3` is factor of `x+2x^(3)-9x^(2)+12`

Explanation

(i) Let `p(x) =x^(3)-x^(2)+11 x+69`
we have to show that, `x+3` is a factor of `p(x)` .
`i.e., p(-3)=0`
Now, `p(-3)=(-3)^(3)-(-3)^(2)+11(-3)+69`
`=-2-9-33+69=-69+69=0`
hence, `(x+3)` is a factor of `p(x)` .
(ii) Let `p(x) =2x^(3)-9x^(2)+x+12`
we have to show that, `2x -3` is a factor of `p(x).`
i.e., `p((3)/(2))=0`
Now, `p((3)/(2))=2((3)/(2))^(3)-9((3)/(2))^(2)+(3)/(2)+12`
`=2xx(27)/(8) -9xx(9)/(4)+(3)/(2)+12`
` =(27)/(4)-(81)/(4) +(3)/(2)+12`
`=(27-81+6+48)/(4)=(81-81)/(4)=0`
Hence, `(2x-3)` is a factor of `p(x)`.