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Cubic coefficients (Posted on 2020-11-02) Difficulty: 2 of 5
The coefficients a,b,c of a polynomial f:R->R, f(x)=x3+ax2+bx+c are mutually distinct integers and different from zero. Furthermore, f(a)=a3 and f(b)=b3. Determine a,b and c.

No Solution Yet Submitted by Danish Ahmed Khan    
Rating: 3.0000 (1 votes)

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Some Thoughts Possible solution; not a proof | Comment 1 of 3
I think there may be no solution based on the following.

f(x) = x^3 + ax^2 + bx + c
f(a) = a^3 + a^3 + ab + c = a^3
a^3 + ab + c = 0
f(b) = b^3 + ab^2 + b^2 + c = b^3
ab^2 + b^2 + c = 0
a^3 + ab = ab^2 + b^2
a^3 + ab - ab^2 - b^2 = 0  which works when a=b, but the coefficients have to be distinct non-zero integers.
Just checking with a spreadsheet, it looks like there are no other solutions to this last equation other than a=b.
  Posted by Larry on 2020-11-02 13:31:59
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