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) 3x^{2} + 6 = 3(x^{2} + 2)
(ii) x^{2} – 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 x^{2} + 2 are factors of 3x^{2} + 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 a^{2} – b^{2} = (a + b)(a – b) in Factorization
Factorization of Expressions of the Form a^{2} – b^{2} (different of two squares)
We know, (a + b)(a – b) = a^{2} – b^{2}. So, a^{2} – b^{2} = (a + b)(a – b).
Solved Examples on Introduction to Factorization:
1. Factorize: 25a^{2} – 81b^{2}.
Solution:
25a^{2} – 81b^{2}
= (5a)^{2} – (9b)^{2}
= (5a + 9b)(5a – 9b).
2. 16x^{2} – y^{2}
Solution:
16x^{2} – y^{2}
= (4x)^{2} – y^{2}
= (4x + y)(4x – y)
3. a^{2}x^{2} – b^{2}y^{2}
Solution:
a^{2}x^{2} – b^{2}y^{2}
= (ax)^{2} – (by)^{2}
= (ax + by)(ax – by)
4. 2x^{2} – 18
Solution:
2x^{2} – 18
= 2(x^{2} – 9)
= 2(x^{2} – 3^{2})
= 2(x + 3)(x – 3)
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