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Algebra Difficulty 4.6 AIME Find the answer

Let a,ba, b, and cc be the 3 roots of x3−x+1=0x^{3}-x+1=0. Find 1a+1+1b+1+1c+1\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}.

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

Solution

We can substitute x=y−1x=y-1 to obtain a polynomial having roots a+1,b+1,c+1a+1, b+1, c+1, namely, (y−1)3−(y−1)+1=y3−3y2+2y+1(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, 2−1=−2\frac{2}{-1}=-2.

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