Originally Posted by

**Harry1W** I've been working through Chapter 1 of D. L. Johnson's "Elements of Logic via Numbers and Sets" and have become stumped on the following problem, amongst others.

"Prove that the polynomial $\displaystyle f(x) = x^4 + 2x^2 + 2x + 1998 $ cannot be written as a product of two quadratic polynomials with integer coefficients."

The preceding section (and indeed the other questions in this set) was concerned with 'proof by contradiction', so I assume that I am intended to solve this via this method.

Therefore, for contradiction, assume that $\displaystyle f(x) $ can be written as previously described, i.e.

$\displaystyle f(x) = x^4 + 2x^2 + 2x + 1998 = (Ax^2 + Bx +C)(Dx^2 + Ex + F) $

where $\displaystyle A, B, C, E $ and $\displaystyle F$ are integers. Thus,

$\displaystyle x^4 + 2x^2 + 2x+1998 = AD x^4 + (AE + BD) x^3 + (AF + BE + CD) x^2 + (BF + CE) x + CF $,

and by comparing coefficients we see that:

$\displaystyle \begin{array}{rcl}

A D & = & 4 \\

A E + B D & = & 2 \\

A F + B E + C D & = & 2 \\

C F & = & 1998

\end{array} $.

I suppose for the purposes of the proof then, I should aim to show that there are no integer solutions to these set of simultaneous equations - but I have no idea about how to approach this. Any suggestions would be much appreciated.