Question

A polynomial ax^3+bx^2+cx+d leaves a remainder 10x+11 upon division by x^2-1 and also leaves a remainder -6 upon division by x^2+x+1. What are the values of a,b,c,d?

Answer 1

If you compute Modulo (x^2-1), then you have:

x^2 = 1

x^3 = x

So, we have:

ax^3+bx^2+cx+d =

(a + c) x + b + d

Equating this to 10x+11 for all x, gives:

a + c = 10

b + d = 11

Modulo (x^2+x+1), we have:

x^2 = -(x+1)

x^3 = -x(x+1) = -x^2 - x = 1

So, we have:

ax^3+bx^2+cx+d =

(c-b) x + a - b + d

Equating this to -6 for all x gives:

c-b = 0

a - b + d = -6

Solving the 4 equations for the 4 unknowns gives:

a = 1

b = 9

c = 9

d = 2