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Integer coefficients (Posted on 2007-03-13) Difficulty: 2 of 5
Determine if it is possible that p(x)*q(x)= x5+9x+1, if p(x) and q(x) must be polynomials of degree 1 or higher, with integer coefficients.

See The Solution Submitted by K Sengupta    
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Some Thoughts Outline of solution | Comment 2 of 3 |
The degrees of p and q must be either 1 and 4 or 2 and 3.
The given polynomial has one real root which appears irrational which rules out a degree of 1.

So let p(x) = ax^2 + bx + c
and q(x) = dx^3 + ex^2 + fx + g

It is clear that d = a = +/- 1 and c = g = +/- 1

From here one can create a system of equations relating the rest of the variables.  This system, which I dont have time to explain right now, has no solutions.

  Posted by Jer on 2007-03-13 09:34:49
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