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

## 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

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.