given that the expression ax^3-11x^2+bx+2 is exactly divisible by 2x^2-3x-2, find a and b then factorise completly (do not use long division).

Answer 1

(cx-d)(2x^2 - 3x - 2) = ax^3 - 11x^2 + bx + 2
2cx^3 - (3c+2d)x^2 - (2c-3d)x + 2d = ax^3 - 11x^2 + bx + 2

2c = a
3c+2d = 11
2c-3d = -b
2d = 2

so,

d=1
3c+2=11
c=3
-3 = b
a = 6

check:

(3x-1)(2x^2 - 3x - 2) =
6x^3 - 11x^2 - 3x + 2