HMMT February

Algebra Difficulty 4.6 AIME Find the answer

Let $a, b$ , and $c$ be the 3 roots of $x^{3}-x+1=0$ . Find $\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

We can substitute $x=y-1$ to obtain a polynomial having roots $a+1, b+1, c+1$ , namely, $(y-1)^{3}-(y-1)+1=y^{3}-3 y^{2}+2 y+1$ . The sum of the reciprocals of the roots of this polynomial is, by Viete's formulas, $\frac{2}{-1}=-2$ .

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