# CLASS-8FACTORIZATION - GROUPING TERMS

GROUPING TERMS

Some algebraic expressions can be factorized by rearranging the terms suitable in pairs such that-

1) The terms in each pair have a common factor and

2) when this common factor is taken out, the same expression is left in each pair.

Example.  ax + ay + bx + by

We can regroup  ax + ay + bx + by as  (ax + ay) + (bx + by)

Then, we have to taking out the common factor in each pair,

ax + ay + bx + by = a (x+y) + b (x+y)

so, (x+y) is a factor common to both parts of the expression.

Taking (x+y) out, ax + ay + bx + by = a (x+y) + b (x+y) = (x+y)(a+b)

Another possible way of grouping the terms of the given expression in order to carry out factorization is as follows –

ax + ay + bx + by = (ax + bx) + (ay + by) = x (a+b) + y (a+b) = (a+b) (x+y)

there is more than one possible way of grouping the terms of an expression.

There are some examples are given below for your better understanding –

Example.1)  Factorize x²+ 5y + 5x + xy

Ans.)  x²+ 5y + 5x + xy

= (x²+ 5x) + (5y + xy) = x (x + 5) + y (x + 5) = (x + 5) (x + y)

Alternative way

x²+ 5y + 5x + xy

= (x²+ xy) + (5x + 5y) = x (x + y) + 5 (x + y) =  (x + y) (x + 5)   (Ans.)

Example.2)  Factorize 2ax + 2ay + 3bx + 3by

Ans.)    2ax + 2ay + 3bx + 3by

= 2a (x+y) + 3b (x+y) = (x+y) (2a+3b)

Alternatively

2ax + 2ay + 3bx + 3by

=  (2ax + 3bx) + (2ay + 3by)

=  x (2a+3b) + y (2a+3b) = (2a+3b) (x+y)      (Ans.)

Example.3) Factorize 2ab²+ 2b²x + a + 4bx + 4ab + x

Ans.)       2ab² + 2b²x + a + 4bx + 4ab + x

=  (2ab² + 2b²x) + (a + x) + (4bx + 4ab)

=   2b²(a+x) + (a+x) + 4b(a+x)

=   (a+x)(2b²+1+4b)

Alternatively

2ab² + 2b²x + a + 4bx + 4ab + x

=    (2b²a + a + 4ab) + (2b²x + x + 4bx)

=    a (2b²+1+4b) + x (2b²+1+4b)

=     (2b²+1+4b) (a+x)                    (Ans.)