View source

== 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?

<math>\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</math>

[[Image:2000_12_AMC-22.png]]

== 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>
# 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 sum of the coefficients is <math>P(1) > 1.5</math>
# The sum of the real roots is <math>> 5</math>.

Clearly <math>\mathrm{(C)}</math> is the smallest.

== See also ==
{{AMC12 box|year=2000|num-b=21|num-a=23}}

[[Category:Introductory Algebra Problems]]