If
is a multiple of
, then we can write
-
![{\displaystyle {}P=(X-a)Q\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cbf4e1588c7dc3b12dca9149d80b6dd41950a4a9)
with another polynomial
. Inserting
yields
-
![{\displaystyle {}P(a)=(a-a)Q(a)=0\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/38abd5c4bbc0f6eb22b6a75a68396c7187db5f06)
In general, there exists, due to
fact,
a representation
-
![{\displaystyle {}P=(X-a)Q+R\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7754cf376a03ce78c17c631ad16795d3471d6dd9)
where either
or the degree of
is
, so in both cases
is a constant. Inserting
yields
-
![{\displaystyle {}P(a)=R\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f560e2ca993d262b8326b4b79de92597dfc8e8e)
So if
holds, then the remainder must be
,
and this means
.