Difference between revisions of "2000 AMC 12 Problems/Problem 22"
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== Problem == | == Problem == | ||
The [[graph]] below shows a portion of the [[curve]] defined by the quartic [[polynomial]] <math>P(x) = x^4 + ax^3 + bx^2 + cx + d</math>. Which of the following is the smallest? | The [[graph]] below shows a portion of the [[curve]] defined by the quartic [[polynomial]] <math>P(x) = x^4 + ax^3 + bx^2 + cx + d</math>. Which of the following is the smallest? | ||
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+ | [[Image:2000_12_AMC-22.png]] | ||
<math>\text{(A)}\ P(-1)\\ | <math>\text{(A)}\ P(-1)\\ | ||
\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> |
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== Solution == | == Solution == | ||
− | + | Note that there are 3 maxima/minima. Hence we know that the rest of the graph is greater than 10. We approximate each of the above expressions: | |
# According to the graph, <math>P(-1) > 4</math> | # According to the graph, <math>P(-1) > 4</math> | ||
# The product of the roots is <math>d</math> by [[Vieta’s formulas]]. Also, <math>d = P(0) > 5</math> according to the graph. | # The product of the roots is <math>d</math> by [[Vieta’s formulas]]. Also, <math>d = P(0) > 5</math> according to the graph. | ||
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Clearly <math>\mathrm{(C)}</math> is the smallest. | Clearly <math>\mathrm{(C)}</math> is the smallest. | ||
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− | == See | + | == See Also == |
− | {{AMC12 box|year=2000|num-b= | + | {{AMC12 box|year=2000|num-b=21|num-a=23}} |
[[Category:Introductory Algebra Problems]] | [[Category:Introductory Algebra Problems]] | ||
+ | {{MAA Notice}} |
Latest revision as of 01:54, 19 October 2020
Problem
The graph below shows a portion of the curve defined by the quartic polynomial . Which of the following is the smallest?
Solution
Note that there are 3 maxima/minima. Hence we know that the rest of the graph is greater than 10. 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 |
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