A polynomial p (x) has a remainder of 4 when divided by (x+1) and a remainder of 7 when divided by (x-2) what will be the remainder when divided by (x+1) (x-2)

Hello:

p (x) = (x+1) g (x) + 4 ... (1)

p (x) = (x-2) h (x) + 7 ... (2)

find : R ... p (x) = (x+1) (x-2) L (x) + R (R: remainder)

multiply (1) by : (x-2) and (2) by : (x+1)

(x-2) p (x) = (x-2) (x+1) g (x) + 4 (x-2) ... (a)

(x+1) p (x) = (x-2) (x+1) h (x) + 7 (x+1) ... (b)

(b) - (a):

(x+1) p (x) - (x-2) p (x) = (x-2) (x+1) h (x) - (x-2) (x+1) g (x) + 7 (x+1) - 4 (x-2)

((x+1) - (x-2)) p (x) = (x-2) (x+1) (h (x) - g (x)) + 3x+15

3p (x) = (x-2) (x+1) (h (x) - g (x)) + 3 (x+5)

p (x) = (x-2) (x+1) ((h (x) - g (x)) / 3) + (x+5)

so : L (x) = ((h (x) - g (x)) / 3)

and : R = x+5 is the remainder when divided p (x) by (x+1) (x-2).