Question

How do you factor $ {{x}^{2}} $ − $ {{y}^{2}} $ ?

Answer 1

Hint:
Observe that there is no common term between $ {{x}^{2}} $ and $ {{y}^{2}} $. In order to factorize it, we may have to introduce some new terms (without changing the overall value) which have something in common with the given terms. Two such terms can be 'xy' and '−xy'. Regroup the terms and separate the common factors wherever possible.

Complete Step by step Solution:
Let us write the given expression $ {{x}^{2}} $ − $ {{y}^{2}} $ as:
= $ {{x}^{2}} $ − $ {{y}^{2}} $ + xy − xy
Let's group some terms, as follows:
= $ \left( {{x}^{2}}-xy \right) $ + $ \left( xy-{{y}^{2}} \right) $
Separating the common factors from both the brackets, we get:
= x(x − y) + y(x − y)
Separating the common factor (x − y) from both the above terms, we get:
= (x − y)(x + y), which is the required factorization.

Note:
Factorization is the process of writing a given expression as a product of two or more expressions. Not every given expression can always be factorized. For example, $ {{x}^{2}} $ + $ {{y}^{2}} $ , $ {{x}^{2}} $ + $ {{y}^{2}} $ + xy and many more; but we can factorize them 'partially' (some products and some sum / difference).
Some common factorizations:
 $ {{a}^{2}} $ + $ {{b}^{2}} $ ± 2ab = $ {{(a\pm b)}^{2}} $
 $ {{a}^{3}} $ ± $ {{b}^{3}} $ ± 3ab(a ± b) = $ {{(a\pm b)}^{3}} $
 $ {{a}^{3}} $ ± $ {{b}^{3}} $ = (a ± b)( $ {{a}^{2}} $ $ \mp $ ab + $ {{b}^{2}} $ )