Question

if a polynomial leaves a remainder of 5 when divided by x-3 and a reminder of -7 when divided by x+1, what is the reminder when the polynomial is divided by x^2-2x-3?

Answer 1

Let's say that p(x) = (x-3)(x+1)q(x) + (ax+b)
That is, it has some linear remainder when divided by x^2-2x-3
We know from the Remainder Theorem that
p(3) = 5
p(-1) = -7
That means that
3a+b = 5
-a+b = -7
So, a=3, b=-4
The desired remainder is 3x-4