2001 AMC 12 Problems/Problem 19

Problem

The polynomial $p(x) = x^3+ax^2+bx+c$ has the property that the average of its zeros, the product of its zeros, and the sum of its coefficients are all equal. The $y$ -intercept of the graph of $y=p(x)$ is 2. What is $b$ ?

$(\mathrm{A})\ -11 \qquad (\mathrm{B})\ -10 \qquad (\mathrm{C})\ -9 \qquad (\mathrm{D})\ 1 \qquad (\mathrm{E})\ 5$

Solution

We are given $c=2$ . So the product of the roots is $-c = -2$ by Vieta's formulas. These also tell us that $\frac{-a}{3}$ is the average of the zeros, so $\frac{-a}3=-2 \implies a = 6$ . We are also given that the sum of the coefficients is $-2$ , so $1+6+b+2 = -2 \implies b=-11$ . So the answer is $\fbox{A}$ .

Video Solution by OmegaLearn

https://youtu.be/czk6OsfrbxQ?t=678

~ pi_is_3.14

Video Solution

https://www.youtube.com/watch?v=qtlXAxj4y2Y

2001 AMC 12 (Problems • Answer Key • Resources)
Preceded by Problem 18	Followed by Problem 20
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

Page

Toolbox

Search

2001 AMC 12 Problems/Problem 19

Contents

Problem

Solution

Video Solution by OmegaLearn

Video Solution

See Also