Introduction to factorization in algebra

Introduction to factorization in algebra

                      In Algebra, factorization is the process of simplifying an algebric equation by factoring out the common factors from all terms. Thus, the expression can be expressed in terms of the product of two or more simple expressions. In some cases, the factorization of algebric equations is done by splitting some terms into two or more terms. For example the factorization of a quadratic equation in algebra is done by splitting the middle term in to two terms.

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                      For example, factorization of the polynomial, x² − 16 factors as (x − 4) (x + 4). In all cases, a product of simpler Expressions is obtained. Here x² – 16 is a second degree polynomial.Those factors  (x + 4) and (x – 4) are first degree polynomials. This factorization is useful in simplifying expressions.

Methods of Factorization:

1) Factorization in algebra by removing common factors:

When the terms of an algebraic expression X  have a common factor Y, we divide each term of X by Y and get an expression Z. Now, X is factored as Y × Z.

Ex:Factorize: 6x⁴y³ – 4x²y² + 10xy³.

Solution : We observe that 2xy² is a common factor .

6x⁴y³ – 4x²y² + 10xy³ =2xy²[ (6x⁴y³)/2xy² )-( 4x²y²/2xy² )+( 10xy³/2xy² ) ]=2xy²(3x³y – 2x + 5y).

2)Factorization in algebra  by grouping method:

Solution:Grouping method is used for the given expression contains three or more terms.

Steps:

In the first step, if the terms having common factors, then those terms to be grouped.

In the second step, the greatest common factor (GCF) is taken out.

Finally, in the third step, the distributive law is used to find the factors.

The distributive law is,

a (b + c)=a b + a c

Factoring in algebra by grouping: x³+3x²−6x−18

Solution:In the first step, if the terms have common factors, then those terms to be grouped.

(x³+3x²) + (−6x−18)

Factor x² out of the first two terms, and factor −6 out of the second two terms.

x²(x+3)-6(x+3)

Note that there is a common factor, x+3. So, take (x+3) as common.

(x+3) (x²-6) is the final factorization.

x³+3x²−6x−18=(x+3) (x²-6).

Few more Methods of Factorization:

3) Factorization of quadratic expression in algebra:

Consider the quadratic expression like 15 – 2x – x².   

Solution: Writing in the standard form,

15 – 2x – x² = –x² – 2x + 15

                      = (–1) (x² + 2x – 15).

Using the splitting method, If we can find two numbers p and q such that

p + q = 2 and p q = -15, then we can get the factors.

So , p + q = 5+(-3)=2 and p*q =5*(-3)= -15

Hence, we get 15 – 2x – x² = (–1) (x² + 2x – 15).

                                                 = (–1) [(x+5) {x + (–3)}]

                                                 = (–1) (x +5)(x – 3)

                                                 = (x + 5) ( 3 – x).

                                                 = (3x + 1) (2x + 5)

4) Factoring Difference of Two Squares:

Factorization of the difference of the two squares in the form x² - y², we should remember the formula

(x²-y²)=(x + y) (x - y).This is the special cases of factorization.

Example: a²-4 = (a + 2) (a - 2)