This is not the document you are looking for? Use the search form below to find more!

Report

# Factoring

Document Description
Factorization refers to factoring a polynomial into irreducible polynomials over a given field. Other factorizations, such as square-free factorization exist, but the irreducible factorization, the most common, is the subject of this article. It depends strongly on the choice of field. For example, the fundamental theorem of algebra, which states that all polynomials with complex coefficients have complex roots, implies that a polynomial with integer coefficients can be completely reduced to linear factors over the complex field C. On the other hand, such a polynomial may only be reducible to linear and quadratic factors over the real field R. Over the rational number field Q, it is possible that no factorization at all may be possible. From a more practical vantage point.
File Details
• File size: 176.21kb
• Pages: 3
• Tags: factoring, polynomial multiplication, multiplying polynomials
• content preview
Submitter
Embed Code:

Related Documents

## RTS Financial Factoring

by: mufida, 51 pages

Meet Travis. Factoring with RTS Financial When he started, Travis had 2 trucks. When he started, Travis had 2 trucks. Now Travis has 10 trucks. Travis uses ...

## Factoring Calculator

by: mahesh4528, 3 pages

Steps for Factoring Step 1 : To find the factors, divide the given number by all natural numbers till the given number. If remainder comes zero, they are factors, else not. Example problems on ...

## Factoring Calculator

by: storysubmission11, 3 pages

Mathematics is a never ending subject which is related to measurement, patterns, of numbers in relation with operators and variables. For providing help for students in math, an online tutoring ...

## Solving Quadratic Equations by Factoring

by: isabelle, 13 pages

Solving Quadratic Equations by FactoringQuadratic Equations are also known as Second Degree Equations because the highest power of the variable is 2. They may have zero, one or ...

by: anssi, 5 pages

Factoring Quadratics In the form of: ax 2 + bx + c Look at the signs of the third term Now look at the signs of the second term POSITIVE…

## Freight Bill Factoring - Getting Paid on Time, Every Time

by: jessiesnyder123, 2 pages

Every industry deals with the same problem: not getting paid on time. This can be a detriment to the family and to the business. When invoices are sent out, that still doesn't guarantee a prompt ...

## Maternal Grandmothers do go the Extra Mile : Factoring Distance and Lineage into Differential Contact with Grandchildren

by: shinta, 12 pages

Several studies conducted from an evolutionary perspective have documented differential investment in grandchildren by lineage. The majority of these studies have used retrospective ...

by: ronja, 13 pages

by: billytrail, 1 pages

Merchant cash advance blog reference regarding unsecured business loans and merchant loans for small businesses. Merchant cash advances are extremely popular in 2011.

## A Freight Management Service Can Streamline Your Shipping Costs and Processes

by: kermitmooney1129, 2 pages

These days every business is trying to find ways to keep costs down and maximize their profit. Of course this has always been a top priority, but with the increased competition and shaky economy, ...

Content Preview
Factoring
Factoring
Factorization refers to factoring a polynomial into irreducible polynomials over a given field. Other
factorizations, such as square-free factorization exist, but the irreducible factorization, the most
common, is the subject of this article. It depends strongly on the choice of field. For example, the
fundamental theorem of algebra, which states that all polynomials with complex coefficients have
complex roots, implies that a polynomial with integer coefficients can be completely reduced to linear
factors over the complex field C. On the other hand, such a polynomial may only be reducible to
linear and quadratic factors over the real field R. Over the rational number field Q, it is possible that
no factorization at all may be possible. From a more practical vantage point.
It can be shown that factoring over Q (the rational numbers) can be reduced to factoring over Z (the
integers). This is a specific example of a more general case -- factoring over a field of fractions can
be reduced to factoring over the corresponding integral domain. This algebraic point goes by the name
of Gauss's lemma. The classic proof, due to Gauss, first factors a polynomial into its content, a rational
number, and its primitive part, a polynomial whose coefficients are pure integers and share no
common divisor among them. Any polynomial with rational coefficients can be factored in this way,
using a content composed of the greatest common divisor of the numerators, and the least common
multiple of the denominators. This factorization is unique.
Know More About :- Polynomial Multiplication

Math.Edurite.com
Page : 1/3

Obtaining linear factors :- All linear factors with rational coefficients can be found using the rational
root test. If the polynomial to be factored is , then all possible linear factors are of the form , where is
an integer factor of and is an integer factor of . All possible combinations of integer factors can be
tested for validity, and each valid one can be factored out using polynomial long division. If the original
polynomial is the product of factors at least which two of which are of degree 2 or higher, this
technique will only provide a partial factorization; otherwise the factorization will be complete. Note
that in the case of a cubic polynomial, if the cubic is factorable at all the rational root test will give a
complete factorization, either into a linear factor and an irreducible quadratic factor, or into three linear
factors.
Factorizing quartics : -Main article: Quartic function#Factorization into quadratics Reducible quartic
(fourth degree) polynomials with no linear factors can be factored into quadratics.
Duplicate factors : - Main article: Root-finding algorithm#Finding multiple roots of polynomials If
two or more factors of a polynomial are identical to each other, a situation resulting in multiple roots,
then one can exploit the fact that the duplicated factor will also be a factor of the polynomial's
derivative, which itself is a polynomial of one lower degree. The duplicated factor(s) can be found by
using the Euclidean algorithm to find the greatest common factor of the original polynomial and its
derivative.
Kronecker's method :- Since integer polynomials must factor into integer polynomial factors, and
evaluating integer polynomials at integer values must produce integers, the integer values of a
polynomial can be factored in only a finite number of ways, and produce only a finite number of
possible polynomial factors. If this polynomial factors over Z, then at least one of its factors must be of
degree two or less. We need three values to uniquely fit a second degree polynomial. We'll use , and .
Now, 2 can only factor as 1x2, 2x1, (-1)x(-2), or (-2)x(-1).
possible combinations, of which half can be discarded as the negatives of the other half, corresponding
to 64 possible second degree integer polynomials which must be checked. These are the only possible
integer polynomial factors of . Testing them exhaustively reveals. constructed from , and , factors .

Math.Edurite.com
Page : 2/3

ThankYou
Math.Edurite.Com

# Document Outline

• ﾿

Factoring

Share Factoring to:

example:

http://myblog.wordpress.com/
or
http://myblog.com/

Share Factoring as:

From:

To:

Share Factoring.

Enter two words as shown below. If you cannot read the words, click the refresh icon.

Share Factoring as:

Copy html code above and paste to your web page.