Question

what must be added to x cube minus 3 x square - 12 x + 19 so that the result is exactly divisible by X square + X - 6

Answer 1

Given x^3 - 3x^2 - 12x + 19 is divisible by x^2 + x  - 6.

x^2  + x - 6)  x^3 - 3x^2 - 12x + 19(x - 4

                     x^3  + x^2  - 6x
                    ----------------------------------
                                4x^2 - 6x + 19

                                4x^2 - 4x + 24
                   ------------------------------------
                                          -2x - 5 (or) 2x + 5


Hence 2x + 5 must be added.