2008 iTest Problems/Problem 53

Problem

Find the sum of the $2007$ roots of $(x-1)^{2007}+2(x-2)^{2006}+3(x-3)^{2005}+\cdots+2006(x-2006)^2+2007(x-2007)$ .

Solution

Because of Vieta's Formulas, if we know the coefficient of the $x^{2007}$ and $x^{2006}$ term, we can find the sum of all the roots. The coefficient of the $x^{2007}$ term is easy to find -- it's $1$ . Using the Binomial Theorem in $(x-1)^{2007}$ , the coefficient of the $x^{2006}$ term is $-\tbinom{2007}{2006} + 2 = -2005$ . Thus, by Vieta's Formulas, the sum of all $2007$ roots is $\tfrac{-(-2005)}{1} = \boxed{2005}$ .

2008 iTest (Problems)
Preceded by: Problem 52	Followed by: Problem 54
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2008 iTest Problems/Problem 53

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