When x3-2x2+ax=b is divided by x2-2x-3, the remainder is x-6. The value of a and b respectively
Let p(x) = x3 -2(x2) + ax - b
q (x) = x2 -2x -3
r (x) = x - 6
Therefore,
f(x) = p (x) – r (x)
f(x) = x3 - 2x2 + ax - b - x - 6
= x3- 2x2 + (a - 1) x - (b - 6)
q(x) = x2 - 2x - 3
= (x + 1) (x - 3)
Thus,
(x + 1) and (x - 3) are factor of f (x)
a + b = 4
f (3) = 0
33 - 2 (3)2 + (a-1) 3 - b + 6 = 0
12 + 3a - b = 0
a = - 2, b = 6