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?
The ages are integers that obey 7<b<c, so c>=9.
Then f(c)=0 implies (x-c) is a factor of f(x). Let's say f(x)=g(x)*(x-c).
Now evaluate this at x=7. Then 77=g(7)*(7-c)
7-c is a negative integer, and since we are working over integers then 7-c must also be a negative factor of 77: -1, -7, -11, -77.
7-c=-1 implies c=8, but we need c>=9. So this case is rejected.
7-c=-7 implies c=14. Sensible for an oldest child
7-c=-11 implies c=18. Still plausible for an oldest child
7-c=-77 implies c=84. This is far too large for a child when they are to have a 7 year old sibling, so this case is rejected.
So at this point c=14 or c=18.
Case 1, c=14.
b is limited to an integer 8 to 13
Then evaluate f(b) to get 85 = g(b)*(14-b). Again we are working over integers. 14-b will be a positive factor of 85, but also 14-b is limited to an integer 1 to 6. Then the two factors of 85 which are at most 6 are 1 and 5. Then 14-b=1 or 14-b=5, which makes b=13 or b=9.
At this point I will introduce a simplifying restriction: the polynomial is a quadratic which is the smallest order to guarantee to pass through three points.
If f(7)=77, f(13)=85, and f(14)=0 then f(x) = -3x^2+52x-140. This fits all the problem conditions and the ages of the children are 7, 9, and 14.
If f(7)=77, f(13)=85, and f(14)=0 then f(x) = -(37/3)x^2+248x-3162/3, but these coefficients are not integers, so must be rejected.
I let Wolfram Alpha do the number crunching of curve fitting
Case 2, c=18.
b is limited to an integer 8 to 17. Everything is basically the same logic as in Case 1, except this time we get b=17 or 13.
If f(7)=77, f(13)=85, and f(18)=0 then f(x) = -(5/3)x^2+(104/3)x-84, but these coefficients are not integers, so must be rejected.
If f(7)=77, f(17)=85, and f(18)=0 then f(x) = -(39/5)x^2+188x-4284/5, but these coefficients are not integers, so must be rejected.
Exactly once case pulls through, the case where the ages are 7, 9, and 14.
Solution
