A polynomial f with integer coefficients is written on the blackboard. The teacher is a mathematician who has 3 kids named Andrew, Beth and Charles. Andrew, who is 7, is the youngest, and Charles is the oldest. When evaluating the polynomial on his kids' ages he obtains:

f(7) = 77
f(b) = 85, where b is Beth's age,
f(c) = 0, where c is Charles' age.

How old is each child?