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.)