2000 AMC 12 Problems/Problem 22
Problem
The graph below shows a portion of the curve defined by the quartic polynomial . Which of the following is the smallest?
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Solution
We note that there are no more zeros of this polynomial, as there already have been three turns in the curve. We approximate each of the above expressions:
- According to the graph,
- The product of the roots is by Vieta’s formulas. Also, according to the graph.
- The product of the real roots is , and the total product is (from above), so the product of the non-real roots is .
- The sum of the coefficients is
- The sum of the real roots is .
Clearly is the smallest.
See also
2000 AMC 12 (Problems • Answer Key • Resources) | |
Preceded by Problem 20 |
Followed by Problem 22 |
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All AMC 12 Problems and Solutions |