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Example:Factorize a Polynomial

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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 x2 − 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.




Factorization of polynomials


In mathematics and computer algebra, polynomial factorization refers to factoring a polynomial into irreducible polynomials over a given field.




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)




The above explanation is copied from Wikipedia, the free encyclopedia and is remixed as allowed under the Creative Commons Attribution- ShareAlike 3.0 Unported License.