CLASS-9ALGEBRA -FACTORIZATION BY GROUPING

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