Factorization

A Linear Factorization form of a polynomial is in which each factor is a linear polynomial. Linear functions are functions of the form y = mx + b;  their graphs are straight lines. A related concept is the “linear factor”:

Factoring is the decomposition of an object (for example, a polynomial, or a matrix) into a product of other objects, or factors, which when multiplied together gives the original number.

For example, the number 15 has prime factors as 5× 3, and the polynomial x2 ? 9 factors as

(x ? 3)(x + 3). In all cases, a product of simpler objects is obtained.

The aim of factorization is usually to reduce something to "basic building blocks," For example numbers to prime numbers, or polynomials to irreducible polynomials.

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Definition: Linear factor

words: “linear factor with root r ”
usage: r is a some real number
meaning: the expression (x ? r)


Examples for Linear factorization:

Through the product of linear polynomial factorization and some of their linear factors are,

1) The polynomial f (x) = x3?2x2 ? x + 2 can be written as a product of linear factors,

f (x) = (x + 1)(x ? 1)(x ? 2).

2) The polynomial f (x) = x3 ? 4x2 + 5x ? 2 can be written as a product of linear factors,

F(x) = (x ? 1)2(x ? 2) = (x ? 1)(x ? 1)( x ? 2).

3) The polynomial f (x) = x3 ? 8 can be factored as f (x) = (x2 + 2x + 4)(x ? 2), but only one of these factors is a linear factor. The other factor, (x2 + 2x + 4) cannot be broken down further into linear factors.

Relation between factors and roots of a polynomial:

There is an important correspondence between the roots of a polynomial and its linear factors:

Roots of f  `<=>` factors of f

r is a root of f   `<=>`  (x ? r ) is a factor of f

The function f (x) = x2 ?5x + 6 has roots r = 2 and r = 3 that correspond to the linear factors in the factorization f (x)=(x?2)(x?3) .

The function f (x) = x2 ?5 has roots r = ?5 and r = ??5 that correspond to the linear factors in the factorization f (x) = (x ? ?5) (x + ?5).

The function f (x) = x2 +5 has no roots, and f cannot be factored into the form f (x) = (x ? a)(x ? b).

The function f (x)=x2 +2x+4 has no roots, and f cannot be factored into the form f (x) = (x ? a)(x ? b).

The function f (x) = x3 ? 2x2 ? x + 2 has roots r = ?1, r =1, and r = 2 that correspond to the linear factors in the factorization

f (x) = (x + 1)(x ? 1)(x ? 2).

The function f (x) = x3 ? 4x2 + 5x ? 2 has roots r =1, and r = 2 that correspond to the linear factors in the factorization

f(x) = (x ? 1)2( x ? 2) = (x ? 1)( x ? 1)( x ? 2).


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The function f (x) = x3 ? 8 has only one root, r = 2. We have seen that f can be factored f(x) = (x2 + 2x + 4) (x ? 2), but f cannot be broken down completely into linear factors. The linear factor (x ? 2) corresponds to the root r = 2, but the factor (x2 + 2x + 4) is not a linear factor and does not correspond to any roots.