Explanation:
Factorization
In mathematics, factorization (also factorisation in British English) or factoring is the decomposition of an object (for
example, a number, a polynomial,
or a matrix) into a product of other objects, or factors, which when multiplied together give the original. For
example, the number 15 factors into primes as 3 × 5, and the polynomial x^{2} − 4 factors as (x − 2) (x + 2). In all cases, a product of
simpler objects is obtained.
The aim of factoring is usually to reduce something to "basic
building blocks," such as numbers to prime numbers, or polynomials to irreducible
polynomials.
http://en.wikipedia.org/wiki/Factorization#Polynomials
Factorization of polynomials
In mathematics and computer algebra, polynomial factorization refers to factoring a
polynomial into irreducible polynomials over a given field.
http://en.wikipedia.org/wiki/Factorization_of_polynomials
Factor theorem
In algebra, the factor theorem is a theorem linking factors and zeros of a polynomial.
It is a special case of the polynomial remainder theorem.
The factor theorem states that a polynomial f(x) has a factor (x − k) if and only if f (k)
= 0.
Factorization
of polynomials
Two problems where the factor theorem is commonly applied are
those of factoring a polynomial and finding the roots of a polynomial equation;
it is a direct consequence of the theorem that these problems are essentially
equivalent.
The factor theorem is also used to remove known zeros from a
polynomial while leaving all unknown zeros intact, thus producing a lower
degree polynomial whose zeros may be easier to find. Abstractly, the method is
as follows:
1. "Guess"
a zero a of
the polynomial f. (In general, this can be very hard, but math textbook
problems that involve solving a polynomial equation are often designed so that
some roots are easy to discover.)
2. Use
the factor theorem to conclude that (x − a) is a factor of f(x).
3. Compute
the polynomial, for
example using polynomial long
division.
4. Conclude that any root of f(x) = 0 is a root of g(x) = 0. Since the polynomial degree of g is
one less than that of f, it is "simpler" to find the
remaining zeros by studying. (Our solved example in mathguru.com uses
this concept)
http://en.wikipedia.org/wiki/Factor_theorem
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