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: vedadehhc and tdeng