Problem 13 Polynomials
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Problem 13: Polynomials
HMMT 2008 Guts #13
Problem: Let P (x) be a polynomial with degree 2008 and leading coefficient 1 such that P (0) = 2007, P (1) = 2006, P (2) = 2005, . . . , P (2007) = 0. Determine the value of P (2008). Solution: We want to find a way to relate P (x) to a polynomial we know the roots of. We consider polynomial Q(x) = P (x) − 2007 + x. Notice that Q(x) has a root for x = 0, 1, 2, ..., 2007. So we can rewrite Q(x) as Q(x) = x(x − 1)(x − 2)...(x − 2006)(x − 2007). So we know Q(2008) = 2008! and we can just plug in 2008 for x, so we get P (2008) = Q(2008) + 2007 − x = 2008! − 1 .
Solution was written by Sean Soni and compiled from Art of Problem Solving Forums.
HMMT 2008 Guts #13
Problem: Let P (x) be a polynomial with degree 2008 and leading coefficient 1 such that P (0) = 2007, P (1) = 2006, P (2) = 2005, . . . , P (2007) = 0. Determine the value of P (2008). Solution: We want to find a way to relate P (x) to a polynomial we know the roots of. We consider polynomial Q(x) = P (x) − 2007 + x. Notice that Q(x) has a root for x = 0, 1, 2, ..., 2007. So we can rewrite Q(x) as Q(x) = x(x − 1)(x − 2)...(x − 2006)(x − 2007). So we know Q(2008) = 2008! and we can just plug in 2008 for x, so we get P (2008) = Q(2008) + 2007 − x = 2008! − 1 .
Solution was written by Sean Soni and compiled from Art of Problem Solving Forums.
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