ce maths 2002 mc 38

The remainder when x^2 + ax + b is divided by x+2 is -4.

It means that (-2)^2 - 2a +b = -4 or 2a - b = 8 --------(1)

The remainder when ax^2 + bx +1 is divided by x-2 is 9

It means that a(2)^2+2b+1 =9 or 4a+2b=8 or 2a+b=4 ----- (2)

(1) + (2), we have 4a = 12, a=3 (and b= -2)

Let f(x)=x^2 + ax + b
g(x)=ax^2 + bx +1

The remainder when x^2 + ax + b is divided by x+2 is -4
f(-2)= 4-2a+b=-4
-2a+b=-8
2a-b=8 ------------(*)

The remainder when ax^2 + bx +1 is divided by x-2 is 9. The value of a is
g(2) = 4a +2b+1 =9
4a +2b =8
2a +b =4 -------------(**)

(**)-(*) 2b = -4
b=-2
From (**) 2a+(-2)=4
2a=4-(-2)
2a=4+2
2a=6
a=3