2017 AMC 12B Problems/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 ?
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.
No need to find the equations for the lines, really. First of all, . Let's say the line is , and is the coordinate of the third intersection, then , , are the three roots of . Apparently the value of and have no effect on the sum of the 3 roots, because the coefficient of the term is always . So we have, Add them up we have Solve it, we get . .
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