Difference between revisions of "2000 AMC 12 Problems/Problem 22"
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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? | ||
| − | + | [[Image:2000_12_AMC-22.png]] | |
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| − | + | <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. | ||
| − | == See | + | == Video Solution == |
| + | https://youtu.be/MMIEbkGu-k8 | ||
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| + | == See Also == | ||
{{AMC12 box|year=2000|num-b=21|num-a=23}} | {{AMC12 box|year=2000|num-b=21|num-a=23}} | ||
[[Category:Introductory Algebra Problems]] | [[Category:Introductory Algebra Problems]] | ||
| + | {{MAA Notice}} | ||
Latest revision as of 13:37, 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
.
by 
according to the graph.
, and the total product is 
(from above), so the product of the non-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 | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
