Difference between revisions of "2000 AMC 12 Problems/Problem 22"
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\text{(B)}\ \text{The\ product\ of\ the\ zeros\ of\ } P\\ | \text{(B)}\ \text{The\ product\ of\ the\ zeros\ of\ } P\\ | ||
\text{(C)}\ \text{The\ product\ of\ the\ non-real\ zeros\ of\ } P \\ | \text{(C)}\ \text{The\ product\ of\ the\ non-real\ zeros\ of\ } P \\ | ||
− | \text{(D)}\ \text{The\ | + | \text{(D)}\ \text{The\ sum\ of\ the\ coefficients\ of\ } P \\ |
− | \text{(E)}\ \text{The\ | + | \text{(E)}\ \text{The\ sum\ of\ the\ real\ zeros\ of\ } P</math> |
[[Image:2000_12_AMC-22.png]] | [[Image:2000_12_AMC-22.png]] |
Revision as of 20:30, 6 January 2009
Problem
The graph below shows a portion of the curve defined by the quartic polynomial . Which of the following is the smallest?
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 21 |
Followed by Problem 23 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |