2003 AMC 12A Problems/Problem 21
Contents
Problem 21
The graph of the polynomial
has five distinct -intercepts, one of which is at . Which of the following coefficients cannot be zero?
Solution
According to Vieta's Formula, the sum of the roots of a 5th degree polynomial taken 4 at a time is . Calling the roots and letting (our given zero at the origin), the only way to take four of the roots without taking is . Since all of the other products of 4 roots include , they are all equal to 0. And since all of our roots are distinct, none of the terms in can be zero, meaning the entire expression is not zero. Therefore, is a sum of zeros and a non-zero number, meaning it cannot be zero.
Solution 2
Clearly, since (0,0) is an intercept, e must be 0. But if d was 0, would divide the polynomial, which means it would have a double root at 0, which is impossible, since all five roots are distinct.