  Let $c(x)$ be a monic integer polynomial with coefficients $0$ or $1$. Write
$c(x) = a(x) b(x)$ where $a(x)$ and $b(x)$ are monic polynomials with
non-negative real (not necessarily integer) coefficients. The unfair 0--1
polynomial conjecture states that $a(x)$ and $b(x)$ are necessarily integer
polynomials with coefficients $0$ or $1$. Let $a(x)$ be a candidate factor of a
(currently unknown) 0--1 polynomial. We will assume that we know if a
coefficient is $0$, $1$ or strictly between $0$ and $1$, but that we do not
know the precise value of non-integer coefficients. Given this candidate
$a(x)$, this paper gives an algorithm to either find a $b(x)$ and $c(x)$ with
$a(x) b(x) = c(x)$ such that $b(x)$ has non-negative real coefficients and
$c(x)$ has coefficients $0$ or $1$, or (often) shows that no such $c(x)$ and
$b(x)$ exist. Using this algorithm, we consider all candidate factors with
degree less than or equal to 15. With the exception of 975 candidate factors
(out of a possible 7141686 cases), this algorithm shows that there do not exist
$b(x)$ with non-negative real coefficients and $c(x)$ with coefficients $0$ or
$1$ such that $a(x) b(x) = c(x)$.
