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subject: Methods Of Factoring [print this page]

Introduction of factoring methods:
Introduction of factoring methods:

In mathematics, factorization (also called factorization) 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.(Source: From Wikipedia).

In Algebra, Factorization is the process of splitting an algebraic expression into a product of two or more expressions.

Step 1: When the terms of an algebraic expression P have a common factor Q, we divide each term of P by Q and get an expression X. Now, P is factored as P Q.

Step 2: Grouping the corresponding terms in the expression.

Methods of Factoring:

Factorization using grouping method:

Ex : Resolve into factors x8 x2y6.

Sol : Taking x2 as a common factor,

x8 x2y6 = x2 (x6 y6) = x2 [(x3)2 (y3)2]

= x2 (x3 + y3) (x3 y3)

= x2[(x + y) (x2 xy + y2)] [(xy)(x2 + xy + y2)]

[Here, Formula: x3 + y3= (x + y) (x2 xy + y2) and x3 y3 =(xy) (x2 + xy + y2)]

= x2 (x +y) (xy) (x2 + xy + y2) (x2 xy + y2).

Factorization of quadratic expression method:

Ex : Factorize 8p2 + 2p 3.

Sol : Here, we find 8 3 = 24 = 6 4, 6+(4) = 2

By splitting and grouping, we get

8p2 + 2p 3 = 8p2 + 6p 4p 3

= 2p(4p + 3) (1)(4p+3)

= (4p + 3) (2p 1)

Few more Methods of Factoring:

Factization using formulas:

Factorization using X 2 + 2XY + Y 2 (X + Y)2

Ex : Resolve into factors 4x2 + 12xy + 9y2.

Sol : The given expression can be written as follows:

4x2+ 12xy + 9y2 = (2x)2 + 2(2x)(3y) + (3y)2

Setting X = 2x, Y = 3y, the Right hand side is X 2 + 2XY + Y 2 and so it is factored

as (X + Y)2. Hence we get 4x2 + 12xy + 9y2 = (2x + 3y)2.

Factorization using X 2 2XY + Y 2 (X Y)2

Ex : Factorize p2 18pq + 81q2.

Sol : The given polynomial can be written as follows:

p2 18pq + 81q2 = p2 2(p)(9q) + (9q)2

Setting X = p and Y = 9q, the R.H.S. is X 2 2XY + Y 2 and so it is factorized as (X Y)2.

Hence we get p2 18pq + 81q2 = (p 9q)2.

Sum and difference of two squares method:

Factorization using X 2 Y 2 (X + Y) (X Y)

Ex : Factorize 4x4y2 49.

Sol : Since 4x4y2 = (2x2y)2 and 49 = (7)2,

we have 4x4y2 49 = (2x2y)2 (7)2

= (2x2y + 7) (2x2y 7)

by: jeri

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