We consider a question of Byrnes concerning the minimal degree
n
n
of a polynomial with all coefficients in
{
−
1
,
1
}
\{-1,1\}
which has a zero of a given order
m
m
at
x
=
1
x = 1
. For
m
≤
5
m \le 5
, we prove his conjecture that the monic polynomial of this type of minimal degree is given by
∏
k
=
0
m
−
1
(
x
2
k
−
1
)
\prod _{k=0}^{m-1} (x^{2^{k}}-1)
, but we disprove this for
m
≥
6
m \ge 6
. We prove that a polynomial of this type must have
n
≥
e
m
(
1
+
o
(
1
)
)
n \ge e^{\sqrt {m}(1 + o(1))}
, which is in sharp contrast with the situation when one allows coefficients in
{
−
1
,
0
,
1
}
\{-1,0,1\}
. The proofs use simple number theoretic ideas and depend ultimately on the fact that
−
1
≡
1
(
mod
2
)
-1 \equiv 1 \pmod 2
.