Excerpted from Beachy/Blair, Abstract Algebra, 2nd Ed. © 1996

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§ 4.2 Factors

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.
   If d(x) is any nonzero polynomial in I of minimal degree, then

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).
    The greatest common divisor of f(x) and g(x) is denoted by gcd(f(x),g(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. This step reduces the degrees of the polynomials involved, and so repeating the procedure leads to the greatest common divisor of the two polynomials in a finite number of steps.
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


Dividing by p(x)q(x) allows us to write

1 / p(x)q(x) = a(x)/q(x) + b(x)/p(x),

and so

f(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,

where deg(r(x)) < deg(p(x)n). Then we can divide r(x) by p(x)n-1 to obtain

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

where deg(c(x)) < deg(p(x)n-1). This gives us

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

in which deg(b(x)) < deg(p(x)). This can be continued by induction, to obtain

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

in which the numerators b(x),...,t(x) all have lower degree than that of p(x).


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