Factorization by Grouping –
Factorization by grouping is a very methodical, logical and scientific process obtained from 9th-grade math and very funny too. So, we would like to understand about the factorization via group formation. Sometimes in a given expression it is not possible to take out a common factor directly. However, the terms of the given expression are grouped in such a manner that all the terms have a common factor. This can now be factorized as discussed above
Example.1) Factorize => x² + y – xy – x
x² + y – xy – x
= x² – xy – x + y
= (x² - xy) – (x – y)
= x (x – y) – (x – y) [by take out the common factor (x - y)]
= (x – y) (x – 1) (Ans.)
Example.2) Factorize => 6xy - y² + 12xz – 2yz
Ans.) 6xy - y² + 12xz – 2yz
= 6xy + 12xz - y² - 2yz
= 6x (y + 2z) – y (y + 2z) [by take out the common factor (y + 2z)]
= (6x – y) (y + 2z) (Ans.)
Example.3) Factorize => (ax + by)² + (bx – ay)²
Ans.) (ax + by)² + (bx – ay)²
= a²x² + 2abxy + b²y² + b²x²- 2abxy + a²y²
= a²x² + b²y² + b²x² + a²y²
= a² (x² + y²) + b² (x²+ y²) [by take out the common factor (x² + y²)]
= (x² + y²) (a² + b²) (Ans.)
1 3
Example.4) Factorize => x² + ------ - 2 – 3x + ------
x² x
1 3
Ans.) x² + ------ - 2 – 3x + ------
x² x
1 1 3
= x² + ------ - 2 . x². ------ - 3x + ------
x² x² x
1 1
= (x - ------ )² - 3 (x - ------ )
x x
1
[ by take out the common factor (x - ------) ]
x
1 1
= (x - ------) (x - ------- - 3) (Ans.)
x x