Difference between revisions of "2000 AMC 12 Problems/Problem 22"
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[[Image:2000_12_AMC-22.png]] | [[Image:2000_12_AMC-22.png]] | ||
− | <math>\ | + | <math>\textbf{(A)}\ P(-1)\\ |
− | \ | + | \textbf{(B)}\ \text{The\ product\ of\ the\ zeros\ of\ } P\\ |
− | \ | + | \textbf{(C)}\ \text{The\ product\ of\ the\ non-real\ zeros\ of\ } P \\ |
− | \ | + | \textbf{(D)}\ \text{The\ sum\ of\ the\ coefficients\ of\ } P \\ |
− | \ | + | \textbf{(E)}\ \text{The\ sum\ of\ the\ real\ zeros\ of\ } P</math> |
== Solution == | == Solution == | ||
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# 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. | ||
# The product of the real roots is <math>>5</math>, and the total product is <math><6</math> (from above), so the product of the non-real roots is <math>< \frac{6}{5}</math>. | # The product of the real roots is <math>>5</math>, and the total product is <math><6</math> (from above), so the product of the non-real roots is <math>< \frac{6}{5}</math>. | ||
− | # The sum of the coefficients is <math>P(1) > | + | # The sum of the coefficients is <math>P(1) > 2.5</math> |
# The sum of the real roots is <math>> 5</math>. | # The sum of the real roots is <math>> 5</math>. | ||
Clearly <math>\mathrm{(C)}</math> is the smallest. | Clearly <math>\mathrm{(C)}</math> is the smallest. | ||
+ | |||
+ | == Video Solution == | ||
+ | https://youtu.be/MMIEbkGu-k8 | ||
== See Also == | == See Also == |
Latest revision as of 14:37, 5 June 2022
Contents
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.
Video Solution
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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