HMMT 二月 2009 · 代数 · 第 4 题

英文解析

4. [ 4 ] Suppose a , b and c are integers such that the greatest common divisor of x + ax + b and x + bx + cis x + 1 (in the ring of polynomials in x with integer coefficients), and the least common multiple of
   2 2 3 2
   x + ax + b and x + bx + c is x − 4 x + x + 6. Find a + b + c .
   Answer: − 6
   2 2
   Solution: Since x +1 divides x + ax + b and the constant term is b , we have x + ax + b = ( x +1)( x + b ),
   and similarly x + bx + c = ( x + 1)( x + c ). Therefore, a = b + 1 = c + 2. Furthermore, the least common 2
   3 2
   multiple of the two polynomials is ( x + 1)( x + b )( x + b − 1) = x − 4 x + x + 6, so b = − 2. Thus a = − 1
   and c = − 3, and a + b + c = − 6.
   1 1 1 3