Introduction to Factorization

We will discuss here about the introduction to factorization.

The method of expressing a given polynomial as a product of two or more polynomials is called factorization.

The polynomials whose product is the given polynomial are called its factors.

You are already familiar with simple factorizations of the following types.

(i) 3x2 + 6 = 3(x2 + 2)

(ii) x2 – 7x = x(x – 7)

(iii) ab + ac + xb + xc = a(b + c) + x(b + c) = (b + c)(a + x), [Factorization by grouping]

In (i), 3 and x2 + 2 are factors of 3x2 + 6; in (ii), x  and x – 7 are factors; in (iii) b + c and a + x are factors of ab + ac + xb + xc.

Use of a2 – b2 = (a + b)(a – b) in Factorization

Factorization of Expressions of the Form a2 – b2 (different of two squares)

We know, (a + b)(a – b) = a2 – b2. So, a2 – b2 = (a + b)(a – b).

Solved Examples on Introduction to Factorization:

1. Factorize: 25a2 – 81b2.

Solution:

25a2 – 81b2

= (5a)2 – (9b)2

= (5a + 9b)(5a – 9b).

2. 16x2 – y2

Solution:

16x2 – y2

= (4x)2 – y2

= (4x + y)(4x – y)

3. a2x2 – b2y2

Solution:

a2x2 – b2y2

= (ax)2 – (by)2

= (ax + by)(ax – by)

4. 2x2 – 18

Solution:

2x2 – 18

= 2(x2 – 9)

= 2(x2 – 32)

= 2(x + 3)(x – 3)

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