Question

Let $a,b,c,d$ be positive integers such that $a\ge b\ge c\ge d$. Prove that the equation $x^4+a{x}^3-b{x}^2+cxd=0$ has no integer solution.

Answer 1

Suppose that m is an integer root of $x^4+a{x}^3-b{x}^2+cxd=0$. As $d\ne 0$, we have $m\ne 0$.
Suppose now that m > 0.
Then $m^4+a{m}^3=b{m}^2-cmd>0$ and hence $m>a\ge d$.
If m < 0, then writing $n=m>0$ we have $n^4+a{n}^3-b{n}^2+cnd=n^4+n^2\left(anb\right)+\left(cnd\right)>0$, a contradiction.
This proves that the given polynomial has no integer roots