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)\\
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\text{(D)}\ \text{The\ sum\ of\ the\ coefficients\ of\ } P \\
 
\text{(D)}\ \text{The\ sum\ of\ the\ coefficients\ of\ } P \\
 
\text{(E)}\ \text{The\ sum\ of\ the\ real\ zeros\ of\ } P</math>
 
\text{(E)}\ \text{The\ sum\ of\ the\ real\ zeros\ of\ } P</math>
 
[[Image:2000_12_AMC-22.png]]
 
  
 
== Solution ==
 
== Solution ==
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Clearly <math>\mathrm{(C)}</math> is the smallest.
 
Clearly <math>\mathrm{(C)}</math> is the smallest.
  
== See also ==
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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}}
 
{{MAA Notice}}

Latest revision as of 00:54, 19 October 2020

Problem

The graph below shows a portion of the curve defined by the quartic polynomial $P(x) = x^4 + ax^3 + bx^2 + cx + d$. Which of the following is the smallest?

2000 12 AMC-22.png

$\text{(A)}\ P(-1)\\ \text{(B)}\ \text{The\ product\ of\ the\ zeros\ of\ } P\\ \text{(C)}\ \text{The\ product\ of\ the\ non-real\ zeros\ of\ } P \\ \text{(D)}\ \text{The\ sum\ of\ the\ coefficients\ of\ } P \\ \text{(E)}\ \text{The\ sum\ of\ the\ real\ zeros\ of\ } P$

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:

  1. According to the graph, $P(-1) > 4$
  2. The product of the roots is $d$ by Vieta’s formulas. Also, $d = P(0) > 5$ according to the graph.
  3. The product of the real roots is $>5$, and the total product is $<6$ (from above), so the product of the non-real roots is $< \frac{6}{5}$.
  4. The sum of the coefficients is $P(1) > 1.5$
  5. The sum of the real roots is $> 5$.

Clearly $\mathrm{(C)}$ is the smallest.

See Also

2000 AMC 12 (ProblemsAnswer KeyResources)
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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