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How many cubic (i.e., third-degree) polynomials f(x) are there such that f(x) has nonnegative integer coefficients and f(1) = \(4\)?

 Jan 21, 2022
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f(x)=ax^3+bx^2+cx+d

f(1)=a+b+c+d = 4

a+b+c+d=4

 

If a = 4 then b,c, and d = 0                                               1  possibility

 

If a=3 then one of b,c,d is 1 and the others are 0             3  possibility

 

If a=2 and one of the others is 2                                       3  possibility 

If a=2 and two of the othes are 1                                      3  possibility

 

If a=1 and the other three are all 1                                   1  possibility

If a=1 and one of the others is 3                                       3  possibility

f a=1 and 2 of the others are 1 and 2 respectively          3!=6  possibility

 

Total  1+3+3+3+1+3+6 = 20

 

You need to check that I haven't missed any.

 Jan 21, 2022

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