Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 5.1 AIME, harder Find the answer

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

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

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

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