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Factorisation by Taking Out Common Factors

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Factorisation by Taking Out Common Factors:

A systematic way of factorising an expression is the common factor method.

It consists of three steps:

  1. Write each term of the expression as a product of irreducible factors.
  2. Look for and separate the common factors and
  3. Combine the remaining factors in each term in accordance with the distributive law.

Factorise 5xy + 10x.

The irreducible factor forms of 5xy and 10x are respectively,
5xy = 5 × x × y
10x = 2 × 5 × x
The two terms have 5 and x as common factors. Now,
5xy + 10x = (5 × x × y) + (5 × x × 2)
= (5x × y) + (5x × 2)
We combine the two terms using the distributive law,
(5x × y) + (5x × 2) = 5x ×(y + 2)
Therefore, 5xy + 10x = 5x(y + 2).

Example

Factorise 12a2b+ 15ab2
We have
12a2b = 2 × 2 × 3 × a × a × b.
15ab2 = 3 × 5 × a × b × b.
The two terms have 3, a, and b as common factors.
Therefore,
12a2b + 15ab2 
= (3 × a × b × 2 × 2 × a ) + (3 × a × b × 5 × b)
= 3 × a × b × [(2 × 2 × a) + (5 × b)]        ........(combining the terms)
= 3ab × (4a + 5b)
= 3ab(4a + 5b)                                        .........(required factor form)

Example

Factorise 10x2 - 18x3+ 14x4
`10x^2 = 2 xx 5 xx x xx x`
`18x^3 = 2 xx 3 xx 3 xx x xx x xx x`
`14x^4 = 2 xx 7 xx x xx x xx x xx x`
The common factors of the three terms are 2, x and x.
Therefore,
10x2 - 18x3 + 14x4
`= (2 xx x xx x xx 5) – (2 xx x xx x xx 3 xx 3 xx x) + (2 xx x xx x xx 7 xx x xx x)`
`= 2 xx x xx x xx[(5 – (3 xx 3 xx x) + (7 xx x xx x)]`  .... (combining the three terms) 
`=2x^2 xx (5 –  9x + 7x^2)`
`=2x^2 xx (7x^2 - 9x + 5)`.
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