Difference between revisions of "2000 AMC 12 Problems/Problem 22"

Revision as of 19:30, 6 January 2009

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?

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

Solution

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

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

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

@@ Line 5: / Line 5: @@
 \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\ product\ of\ the\ coefficients\ of\ } P \\
+\text{(D)}\ \text{The\ sum\ of\ the\ coefficients\ of\ } P \\
-\text{(E)}\ \text{The\ product\ 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]]

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Difference between revisions of "2000 AMC 12 Problems/Problem 22"

Revision as of 19:30, 6 January 2009

Problem

Solution

See also