jmc

Question

Suppose a, b and c are integers such that the greatest common divisor of x^2+ax+b and x^2+bx+c is x+1 (in the set of polynomials in x with integer coefficients), and the least common multiple of x^2+ax+b and x^2+bx+c is x^3-4x^2+x+6. Find a+b+c.

Accepted Answer

Since x+1 divides x^2+ax+b and the constant term is b, we have x^2+ax+b=(x+1)(x+b), and similarly x^2+bx+c=(x+1)(x+c). Therefore, a=b+1=c+2. Furthermore, the least common multiple of the two polynomials is (x+1)(x+b)(x+b-1)=x^3-4x^2+x+6, so b=-2. Thus a=-1 and c=-3, and a+b+c=-6.