Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 4.7 AIME Find the answer

Let $f(x)=x^{4}+a x^{3}+b x^{2}+c x+d$ be a polynomial whose roots are all negative integers. If $a+b+c+d=2009$ , find $d$ .

Your answer

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

Solution

Call the roots $-x_{1},-x_{2},-x_{3}$ , and $-x_{4}$ . Then $f(x)$ must factor as $(x+x_{1})(x+x_{2})(x+x_{3})(x+x_{4})$ . If we evaluate $f$ at 1, we get $(1+x_{1})(1+x_{2})(1+x_{3})(1+x_{4})=a+b+c+d+1=2010.2010=2 \cdot 3 \cdot 5 \cdot 67$ . $d$ is the product of the four roots, so $d=(-1) \cdot(-2) \cdot(-4) \cdot(-66)$ .

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