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**Theorem 4.2.1. [Division Algorithm]**
For any polynomials f(x) and g(x) in F[x],
with g(x) 0,
there exist unique polynomials
q(x), r(x) in F[x] such that

f(x) = q(x)g(x) + r(x),

where either deg(r(x)) < deg(g(x)) or r(x) = 0.

**Theorem 4.2.2**
Let I be a subset of F[x] that satisfies the following conditions:

**(i)**I contains a nonzero polynomial;**(ii)**if f(x), g(x) belong to I, then f(x)+g(x) belongs to I;**(iii)**if f(x) belongs to I and q(x) is any polynomial in F[x], then q(x)f(x) belongs to I.

I = { f(x) in F[x] | f(x) = q(x)d(x) for some q(x) in F[x] }.

**Definition 4.2.3.**
A monic polynomial d(x) in F[x] is called the
** greatest common divisor**
of f(x), g(x) in F[x] if

**(i)**d(x) | f(x) and d(x) | g(x) , and**(ii)**if h(x) | f(x) and h(x) | g(x) for some h(x) in F[x], then h(x) | d(x).

If gcd(f(x),g(x)) = 1, then the polynomials f(x) and g(x) are said to be
** relatively prime**.

**Theorem 4.2.4.**
For any nonzero polynomials f(x),g(x) in F[x],
the greatest common divisor gcd(f(x),g(x)) exists and can be expressed
as a linear combination of f(x) and g(x), in the form

gcd(f(x),g(x)) = a(x)f(x) + b(x)g(x)

for some a(x), b(x) in F[x].

**Example.** 4.2.3.
(Euclidean algorithm for polynomials)
Let f(x), g(x) be nonzero polynomials in F[x].
We can use the division algorithm to write

f(x) = q(x)g(x) + r(x),

with deg(r(x))<deg(g(x)) or r(x) = 0.- If r(x) = 0, then g(x) is a divisor of f(x),
and so gcd(f(x),g(x)) = cg(x), for some c in F.
- If r(x) 0, then it is easy to check that gcd(f(x),g(x)) = gcd(g(x),r(x)).

The Euclidean algorithm for polynomials is similar to the Euclidean algorithm for finding the greatest common divisor of nonzero integers. The polynomials a(x) and b(x) for which

gcd(f(x),g(x)) = a(x)f(x) + b(x)g(x)

can be found just as for integers (see the Euclidean algorithm for integers).

**Proposition 4.2.5.**
Let p(x),f(x),g(x) be polynomials in F[x].
If gcd(p(x),f(x)) = 1 and p(x) | f(x)g(x), then p(x) | g(x).

**Definition 4.2.6.**
A nonconstant polynomial is said to be
**irreducible over the field**
F if it cannot be factored in F[x]
into a product of polynomials of lower degree. It is said to be
**reducible**
over F if such a factorization exists.

**Proposition 4.2.7.**
A polynomial of degree 2 or 3 is irreducible over the field F
if and only if it has no roots in F.

**Lemma 4.2.8.**
The nonconstant polynomial p(x) in F[x]
is irreducible over F if and only if for all f(x), g(x) in F[x],

p(x) | (f(x)g(x)) implies p(x) | f(x) or p(x) | g(x).

**Theorem 4.2.9. [Unique Factorization]**
Any nonconstant polynomial with coefficients in the field F
can be expressed as an element of F times a product of monic polynomials,
each of which is irreducible over the field F .
This expression is unique except for the order in which the factors occur.

**Definition 4.2.10.**
Let f(x) be a polynomial in F[x].
An element c in F is said to be a
**root of multiplicity**
n 1 of f(x) if

(x - c)^{n} | f(x)
but
(x - c)^{n+1}
f(x).

**Proposition 4.2.11.**
A nonconstant polynomial f(x) over the field **R**
of real numbers has no repeated factors
if and only if gcd(f(x),f'(x))=1,
where f'(x) is the derivative of f(x).

**Example.** 4.2.4. (Partial fractions)
Let f(x)/g(x) be a rational function.
The first step in achieving a partial fraction decomposition of f(x)/g(x)
is to use
Theorem 4.2.9
to write g(x) as a product of irreducible polynomials.

If g(x)=p(x)q(x), where p(x) and q(x) are relatively prime, then by
Theorem 4.2.4
there exist polynomials a(x) and b(x) with

a(x)p(x)+b(x)q(x)=1.

Dividing by p(x)q(x) allows us to write1 / p(x)q(x) = a(x)/q(x) + b(x)/p(x),

and sof(x) / g(x) = (f(x)a(x)) / q(x) + (f(x)b(x)) / p(x).

This process can be extended by induction until f(x)/g(x) is written as a sum of rational functions, where in each case the denominator is a power of an irreducible polynomial.
The next step in the partial fraction decomposition
is to expand the terms of the form
h(x)/p(x)^{n}.
Using the division algorithm, we can write

h(x) / p(x)^{n} =
a(x) + r(x)/p(x)^{n},

r(x) = b(x)p(x)^{n-1} + c(x),

r(x) / p(x)^{n} = b(x)/p(x) + c(x)/p(x)^{n},

h(x) / p(x)^{n}=
a(x) + b(x)/p(x) + · · · + t(x)/p(x)^{n},

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