2001 AMC 12 Problems/Problem 19
Contents
Problem
The polynomial has the property that the average of its zeros, the product of its zeros, and the sum of its coefficients are all equal. The
-intercept of the graph of
is 2. What is
?
Solution
We are given . So the product of the roots is
by Vieta's formulas. These also tell us that
is the average of the zeros, so
. We are also given that the sum of the coefficients is
, so
. So the answer is
.
Solution 2 (inefficient)
If you don't know Vieta's formulas, you can go through the desperate route. Let . After distributing, we find that
. We know that the last term (
) is
, so
must be
. We know from our equation that
. The problem gives that the average of the roots is equal to their product (their product is
), so the sum of the roots is hence
(giving us
, since a is the sum of the roots times
). Finally, we know that the sum of the coefficients is the same as the product of the roots, so the sum of the coefficients is
. Thus
, so
, so
and
. The answer is then
.
Video Solution by OmegaLearn
https://youtu.be/czk6OsfrbxQ?t=678
~ pi_is_3.14
Video Solution
https://www.youtube.com/watch?v=qtlXAxj4y2Y
See Also
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 |
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