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Factorisation by Regrouping Terms

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Factorisation by regrouping terms:

Sometimes, all the terms in a given expression do not have a common factor; but the terms can be grouped in such a way that all the terms in each group have a common factor. When we do this, there emerges a common factor across all the groups leading to the required factorisation of the expression. This is the method of regrouping.

In factorisation by regrouping, we should remember that any regrouping (i.e., rearrangement) of the terms in the given expression may not lead to factorisation. We must observe the expression and come out with the desired regrouping by trial and error.

Suppose, the expression was given as 2xy + 3 + 2y + 3x; then it will not be easy to see the factorisation. Rearranging the expression, as 2xy + 2y + 3x + 3, allows us to form groups (2xy + 2y) and (3x + 3) leading to factorisation. This is regrouping. Regrouping may be possible in more than one ways.
Suppose, we regroup the expression as: 2xy + 3x + 2y + 3. This will also lead to factors. Let us try:
2xy + 2y + 3x + 3 
= 2 × x × y + 3 × x + 2 × y + 3
= x × (2y + 3) + 1 × (2y + 3)
= (2y + 3)(x + 1).

Factorise 6xy – 4y + 6 – 9x.

Solution:

Step 1: Check if there is a common factor among all terms. There is none.

Step 2: First two terms have a common factor 2y;

6xy – 4y = 2y(3x – 2)      ...........(1)

If you change their order to – 9x + 6, the factor ( 3x – 2) will come out;

– 9x + 6

= – 3(3x) + 3(2)

= – 3(3x – 2)                  ..........(2)

Step 3: Putting (1) and (2) together,

6xy – 4y + 6 – 9x

= 6xy – 4y – 9x + 6

= 2y(3x – 2) – 3(3x – 2)

= (3x – 2)(2y – 3)

The factors of (6xy – 4y + 6 – 9 x) are (3x – 2) and (2y – 3).

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