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 $P$

Answer: C

Concepts:polynomial Vieta’s Formulas complex number

Difficulty rating: 2030

Solution:

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.

2000 AMC 12 Problem 22

Answer: C

Difficulty rating: 2030

Solution:

Problem 22 in Other Years