1. How many cubic polynomials f(x) are there such that f(x) has nonnegative integer coefficients and f(1) = 9?
2. How many ordered quadruples (a,b,c,d) satisfy, a + b + c + d = 18,
where a,b,c,d are integers such that |a|, |b|, |c|, |d| are each at most 10?
Hint for #1:
cubic polynomial: f(x) = a·x3 + b·x2 + c·x + d
f(1) = a·(1)3 + b·(1)2 + c·(1) + d = 9
f(1) = a + b + c + d = 9
Since each of the coefficients must be a positive whole number, the values for a, b, c, and d must be:
(in some order) 1, 1, 1, 6
1, 1, 2, 5
1, 1, 3, 4
1, 2, 2, 4
1, 2, 3, 3
2, 2, 2, 3
Now, the problem is to find how many different ways each of the above can written.