We study the problem of determining the minimal degree
d
(
m
)
d(m)
of a polynomial that has all coefficients in
{
0
,
1
}
\{0,1\}
and a zero of multiplicity
m
m
at
−
1
-1
. We show that a greedy solution is optimal precisely when
m
≤
5
m\leq 5
, mirroring a result of Boyd on polynomials with
±
1
\pm 1
coefficients. We then examine polynomials of the form
∏
e
∈
E
(
x
e
+
1
)
\prod _{e\in E} (x^e+1)
, where
E
E
is a set of
m
m
positive odd integers with distinct subset sums, and we develop algorithms to determine the minimal degree of such a polynomial. We determine that
d
(
m
)
d(m)
satisfies inequalities of the form
2
m
+
c
1
m
≤
d
(
m
)
≤
103
96
⋅
2
m
+
c
2
2^m + c_1 m \leq d(m) \leq \frac {103}{96}\cdot 2^m + c_2
. Last, we consider the related problem of finding a set of
m
m
positive integers with distinct subset sums and minimal largest element and show that the Conway-Guy sequence yields the optimal solution for
m
≤
9
m\leq 9
, extending some computations of Lunnon.