Question

let R(x) and P(x) be the remainders when  f(x) is divided by (x+2)(x-1) and (x+2)(x-2) respectively. if R(1)= -2, P(2)= 4,find the remainder when f(x) is divided by (x-1)(x-2)

Answer 1

lets for positive constants p and q,
$f(x) = p(x + 2) (x - 1) + R(x) ....1\\ f(x) = q(x + 2) (x - 2) + P(x)....2$

Putting x = 1 in eq.(1) and x = 2 in eq.(2)
f(1) = -2
f(2) = 4

As f(x) is quadratic so lets assume the Remainder on dividing f(x) by (x - 1)(x - 2) is (ax + b), then
$f(x) = \alpha (x - 1) (x - 2) + (ax + b)....3$

Putting x = 1 and x = 2 in eq.(3)
a + b = -2
2a + b = 4

Solving for a and b,
a = 6, b = -8

Answer
6x - 8