Roots of a polynomial
Définition :
Let
be an element of
An element
of
is a root of polynomial
if
Fondamental : Gauss' factorization
If
is a root of
Fondamental : Multiplicity order of a root
Let
be an element of
and
a root of polynomial
From the previous theorem, there exists a polynomial
such that
If
is not a root of
meaning if
we say that
is a simple root (or of order
) of
If
is a root of polynomial
by appliying Gauss' theorem, there exists a polynomial
such that
which gives us:
If
is not a root of polynomial
we say that
is a double root, or of order
of polynomial
If
is a root of polynomial
we go on.
We build this way a sequence of polynomials
with strictly decreasing degrees, hence we end up with:
with
We then say that
is a root of order
of polynomial
Theorem
Let
and
be an element of
is a root of order
of polynomial
if and only if
where
denotes the derivative of order
of polynomial
with the convention
