The Barge Problem: A quintic query for calculus credit

Good news to all those taking calculus this term. Over spring break the Lafayette Mathematics Department implemented a new Extra Credit Policy! Each semester an extra credit problem is to be offered and any student with a correct solution will automatically get an A+ in their calculus course.

The problem for the spring 2018 semester is the following: Among all quintic polynomials

p(x) = (x − a)(x − b)(x − c)(x − d)(x − e)

where a, b, c, d, e are distinct integers, find one so that when p(x) is multiplied out it has the fewest number of nonzero coefficients possible.

Can you earn an A+ in calculus?