2000 AMC 12 #22
Source:
August 26, 2011
algebrapolynomialVieta
Problem Statement
The graph below shows a portion of the curve defined by the quartic polynomial P(x) \equal{} x^4 \plus{} ax^3 \plus{} bx^2 \plus{} cx \plus{} d. Which of the following is the smallest? (A)\ P( \minus{} 1)
(C)\ \text{The product of the non \minus{} real zeros of }P
[asy]
size(170);
defaultpen(linewidth(0.7)+fontsize(7));size(250);
real f(real x) {
real y=1/4;
return 0.2125(x*y)^4-0.625(x*y)^3-1.6125(x*y)^2+0.325(x*y)+5.3;
}
draw(graph(f,-10.5,19.4));
draw((-13,0)--(22,0)^^(0,-10.5)--(0,15));
int i;
filldraw((-13,10.5)--(22,10.5)--(22,20)--(-13,20)--cycle,white, white);
for(i=-3; i<6; i=i+1) {
if(i!=0) {
draw((4*i,0)--(4*i,-0.2));
label(string(i), (4*i,-0.2), S);
}}
for(i=-5; i<6; i=i+1){
if(i!=0) {
draw((0,2*i)--(-0.2,2*i));
label(string(2*i), (-0.2,2*i), W);
}}
label("0", origin, SE);[/asy]