1988 USAMO Problems/Problem 5
Let be the polynomial , where are integers. When expanded in powers of , the coefficient of is and the coefficients of , , ..., are all zero. Find .
First, note that if we reverse the order of the coefficients of each factor, then we will obtain a polynomial whose coefficients are exactly the coefficients of in reverse order. Therefore, if we define the polynomial to be noting that if the polynomial has degree , then the coefficient of is , while the coefficients of for are all .
Let be the sum of the th powers of the roots of . In particular, by Vieta's formulas, we know that . Also, by Newton's Sums, as the coefficients of for are all , we find that Thus for . Now we compute . Note that the roots of are all th roots of unity. If , then the sum of nd powers of these roots will be If , then we can multiply by to obtain But as , this is just . Therefore the sum of the nd powers of the roots of is the same as the sum of the nd powers of the roots of The nd power of each of these roots is just , hence the sum of the nd powers of the roots is On the other hand, we can use the same logic to show that Subtracting (2) from (1) and dividing by 32, we find Therefore, .
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