Look at the expression 2xy + 2y + 3x + 3.
You will notice that the first two terms have common factors 2 and y and the last two terms have a common factor 3. But there is no single favtors common to all the terms.
Suppose (2xy + 2y) in the factor form:
`2xy + 2y = (2 × x × y) + (2 × y)`
`= (2 × y × x) + (2 × y × 1) `
`= (2y × x) + (2y × 1) = 2y (x + 1) `
Similarly , `3x + 3 = (3 × x) + (3 × 1)`
`= 3 × (x + 1) = 3 ( x + 1)`
Hence,` 2xy + 2y + 3x + 3 = 2y (x + 1) + 3 (x +1)`
We observe that a common factors (x + 1) in both the terms on the right hand side.
Combining the two terms,
`2xy + 2y + 3x + 3 = 2y (x + 1) + 3 (x + 1) = (x + 1) (2y + 3) `
The expression 2xy + 2y + 3x + 3 is now in the form of a product of factors. Its factors are (x + 1) and (2y + 3). Note, these factors are irreducible.
In factorization by regrouping sometimes the terms of the given expression need to be arranged in suitable groups in such a way that all the groups have a common factor.
Method of factoring terms:
Step 1: Arrange the terms of the given expression in groups in such a way that all the groups have a common factor.
Step 2: Factorize each group.
Step 3: Take out the factor which is common to each group.