jmc

Question

The polynomial ax^4 + bx^3 + cx^2 + dx + e = 0has coefficients that are all integers, and has roots -2, 5, 9, and -1/3. If e is a positive integer, then find its smallest possible value.

Accepted Answer

By the Integer Root Theorem, -2, 5, and 9 must all divide e, so e must be at least 90. The polynomial (x + 2)(x - 5)(x - 9)(3x + 1) = 3x^4 - 35x^3 + 39x^2 + 287x + 90satisfies the given conditions, so the smallest possible value of e is 90.