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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?

 May 17, 2020
 #1
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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.

 May 17, 2020

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