2000 AMC 12 Problem 22

Question

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?

P(-1)

The product of the zeros of

P

The product of the non-real zeros of

P

The sum of the coefficients of

P

The sum of the real zeros of

P

The graph shows exactly two real zeros, both positive, so

P

has two non-real (conjugate) zeros

The product of all zeros equals

d,

the

y

-intercept; divide it by the product of the real zeros

The graph crosses the

x

-axis exactly twice, both times at positive values, so

P

has two real zeros and two non-real (complex conjugate) zeros.

Reading off the graph: the sum of the coefficients is

P(1) \gt 3;

P(-1) \gt 4;

the sum of the real zeros is greater than

4.5;

and the product of all zeros is

d,

the

y

-intercept, which is less than

6.

Let

R

be the product of the real zeros, so

R \gt 4.5.

The product of the non-real zeros is

\dfrac dR,

which is less than

\dfrac{6}{4.5} \lt 2.

This is smaller than every other listed quantity.

Thus, the correct answer is C.

Problem 22 in other years