<s>
In	O
mathematics	O
,	O
a	O
polynomial	B-Algorithm
decomposition	I-Algorithm
expresses	O
a	O
polynomial	O
f	O
as	O
the	O
functional	B-Application
composition	I-Application
of	O
polynomials	O
g	O
and	O
h	O
,	O
where	O
g	O
and	O
h	O
have	O
degree	O
greater	O
than	O
1	O
;	O
it	O
is	O
an	O
algebraic	O
functional	B-Algorithm
decomposition	I-Algorithm
.	O
</s>
<s>
Polynomials	O
which	O
are	O
decomposable	O
in	O
this	O
way	O
are	O
composite	B-Algorithm
polynomials	I-Algorithm
;	O
those	O
which	O
are	O
not	O
are	O
indecomposable	B-Algorithm
polynomials	I-Algorithm
or	O
sometimes	O
prime	O
polynomials	O
(	O
not	O
to	O
be	O
confused	O
with	O
irreducible	O
polynomials	O
,	O
which	O
cannot	O
be	O
factored	O
into	O
products	O
of	O
polynomials	O
)	O
.	O
</s>
<s>
The	O
degree	O
of	O
a	O
composite	B-Algorithm
polynomial	I-Algorithm
is	O
always	O
a	O
composite	O
number	O
,	O
the	O
product	O
of	O
the	O
degrees	O
of	O
the	O
composed	O
polynomials	O
.	O
</s>
<s>
using	O
the	O
ring	O
operator	O
symbol	O
∘	O
to	O
denote	O
function	B-Application
composition	I-Application
.	O
</s>
<s>
A	O
polynomial	O
may	O
have	O
distinct	O
decompositions	B-Algorithm
into	O
indecomposable	B-Algorithm
polynomials	I-Algorithm
where	O
where	O
for	O
some	O
.	O
</s>
<s>
The	O
restriction	O
in	O
the	O
definition	O
to	O
polynomials	O
of	O
degree	O
greater	O
than	O
one	O
excludes	O
the	O
infinitely	O
many	O
decompositions	B-Algorithm
possible	O
with	O
linear	O
polynomials	O
.	O
</s>
<s>
Joseph	O
Ritt	O
proved	O
that	O
,	O
and	O
the	O
degrees	O
of	O
the	O
components	O
are	O
the	O
same	O
up	O
to	O
linear	O
transformations	O
,	O
but	O
possibly	O
in	O
different	O
order	O
;	O
this	O
is	O
Ritt	O
's	O
polynomial	B-Algorithm
decomposition	I-Algorithm
theorem	O
.	O
</s>
<s>
A	O
polynomial	B-Algorithm
decomposition	I-Algorithm
may	O
enable	O
more	O
efficient	O
evaluation	O
of	O
a	O
polynomial	O
.	O
</s>
<s>
can	O
be	O
calculated	O
with	O
3	O
multiplications	O
and	O
3	O
additions	O
using	O
the	O
decomposition	B-Algorithm
,	O
while	O
Horner	O
's	O
method	O
would	O
require	O
7	O
multiplications	O
and	O
8	O
additions	O
.	O
</s>
<s>
A	O
polynomial	B-Algorithm
decomposition	I-Algorithm
enables	O
calculation	O
of	O
symbolic	O
roots	O
using	O
radicals	O
,	O
even	O
for	O
some	O
irreducible	O
polynomials	O
.	O
</s>
<s>
This	O
technique	O
is	O
used	O
in	O
many	O
computer	B-General_Concept
algebra	I-General_Concept
systems	I-General_Concept
.	O
</s>
<s>
Even	O
in	O
the	O
case	O
of	O
quartic	O
polynomials	O
,	O
where	O
there	O
is	O
an	O
explicit	O
formula	O
for	O
the	O
roots	O
,	O
solving	O
using	O
the	O
decomposition	B-Algorithm
often	O
gives	O
a	O
simpler	O
form	O
.	O
</s>
<s>
The	O
first	O
algorithm	O
for	O
polynomial	B-Algorithm
decomposition	I-Algorithm
was	O
published	O
in	O
1985	O
,	O
though	O
it	O
had	O
been	O
discovered	O
in	O
1976	O
,	O
and	O
implemented	O
in	O
the	O
Macsyma/Maxima	O
computer	O
algebra	O
system	O
.	O
</s>
<s>
A	O
2014	O
algorithm	O
calculates	O
a	O
decomposition	B-Algorithm
in	O
polynomial	O
time	O
and	O
without	O
restrictions	O
on	O
the	O
characteristic	O
.	O
</s>
