2017 AMC 12B Problems/Problem 23
Problem 23
The graph of , where
is a polynomial of degree
, contains points
,
, and
. Lines
,
, and
intersect the graph again at points
,
, and
, respectively, and the sum of the
-coordinates of
,
, and
is 24. What is
?
Solution
First, we can define , which contains points
,
, and
. Now we find that lines
,
, and
are defined by the equations
,
, and
respectively. Since we want to find the
-coordinates of the intersections of these lines and
, we set each of them to
, and synthetically divide by the solutions we already know exist (eg. if we were looking at line
, we would synthetically divide by the solutions
and
, because we already know
intersects the graph at
and
, which have
-coordinates of
and
). After completing this process on all three lines, we get that the
-coordinates of
,
, and
are
,
, and
respectively. Adding these together, we get
which gives us
. Substituting this back into the original equation, we get
, and
Solution by gorefeebuddie
Note: This is a really good AMC 12 problem. It is one of those problems that they have every year.
See Also
2017 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 22 |
Followed by Problem 24 |
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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