2001 AIME I Problems/Problem 3
Problem
Find the sum of the roots, real and non-real, of the equation , given that there are no multiple roots.
Solution 1
From Vieta's formulas, in a polynomial of the form , then the sum of the roots is .
From the Binomial Theorem, the first term of is , but , so the term with the largest degree is . So we need the coefficient of that term, as well as the coefficient of .
Applying Vieta's formulas, we find that the sum of the roots is .
Solution 2
We find that the given equation has a degree polynomial. Note that there are no multiple roots. Thus, if is a root, is also a root. Thus, we pair up pairs of roots that sum to to get a sum of .
Solution 3
Note that if is a root, then is a root and they sum up to We make the substitution so Expanding gives so by Vieta, the sum of the roots of is 0. Since has a degree of 2000, then has 2000 roots so the sum of the roots is
See also
2001 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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