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When a polynomial p(x) is divided by (x+1), the remainder is 4. And when it is divided by (x-2), it's remainder is 3. What is the remainder  when p(x) is divided by x²-x-2

given p(-1) = 4 , p(2) = 3   (remainders) ---------------------(1)

 note that   x²-x-2 = (x+1)(x-2)
let q(x) be the quotient and r(x) = ax + b be the remainder (since you are dividing by a quadratic)
 when p(x) is divided by ( x²-x-2 )

therfore p(x) = (x+1)(x-2)q(x) + ax + b         --------------(2)

putting x = -1  and x = 2  in eqn(2)  and using eqn(1), we get

4 =  -a + b
3= 2a + b

solving,
 a = -1 / 3
 b = 11 /3

therefore remainder = ax + b  = (11- x)/3









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