This paper discusses representations of polynomials that are positive on intervals of the real line. An elementary and constructive proof of the following is given: If
h
(
x
)
,
p
(
x
)
∈
R
[
x
]
h(x), p(x) \in \mathbb {R}[x]
such that
{
α
∈
R
∣
h
(
α
)
≥
0
}
=
[
−
1
,
1
]
\{ \alpha \in \mathbb {R} \mid h(\alpha ) \geq 0 \} = [-1,1]
and
p
(
x
)
>
0
p(x) > 0
on
[
−
1
,
1
]
[-1,1]
, then there exist sums of squares
s
(
x
)
,
t
(
x
)
∈
R
[
x
]
s(x), t(x) \in \mathbb {R}[x]
such that
p
(
x
)
=
s
(
x
)
+
t
(
x
)
h
(
x
)
p(x) = s(x) + t(x) h(x)
. Explicit degree bounds for
s
s
and
t
t
are given, in terms of the degrees of
p
p
and
h
h
and the location of the roots of
p
p
. This is a special case of Schmüdgen’s Theorem, and extends classical results on representations of polynomials positive on a compact interval. Polynomials positive on the non-compact interval
[
0
,
∞
)
[0,\infty )
are also considered.