Reducible polynomial + integer = Reducible polynomial ? As the title says.
More specific :
For every integer $n$, does there exist a pair of polynomials $p(x)$ and $q(x)$ such that:
- $p(x)$ and $q(x)$ each have integer coefficients with no common factor
- $p(x)$ and $q(x)$ are reducible polynomials
- $p(x) + n = q(x)$.
I mean integer nonlinear polynomials without a constant multiplication factor such as $(x^2 + 5)(x+1)$ ( but not $2x^2 $which has constant factor $2$). I think an example clarifies :
$(x)(x-2) + 1 = (x-1)^2$
where $x(x-2)$ and $(x-1)^2$ are the reducible polynomials ( reducible means we can factor them over the ring of integers ) and $+1$ is the integer.
Are there other such identities for the integer $+1$ ? Are there such identities for the integer $+2$ ? ( im aware that $x(x+1) = 0$ mod 2 for integer $x$ but I did not say $x$ is an integer ) How about the integer $+3$ ? Are there such pairs of reducible polynomials for all integers $+n$ ? And if yes , how to find such ?
Edit : I used Owens answer to clarify my question. Edit2 : I ask for additional solutions. Such as higher degree polynomials. Or generalizations. Such as $p(x) + 1 = q(x)$ and $p(x) + 2 = r(x)$.