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Factorisation by Taking Out a Common Factor

When Each Term of the Given Expression Contains a Common Monomial Factor

To factorise this type of algebraic expression, follow these steps:

Step 1: Find the greatest monomial factor (i.e., the HCF) of all the terms.

Step 2: Divide the given expression by the HCF to find the quotient.
Step 3: Write the given expression as a product of the HCF and the quotient.

2 5
3 4
Example 1: Factorise 9x y + 18x y + 27x y .
5 2
3 4

Solution: We have, 9x y = 3 × 3 × x × x × x × y × y × y × y Quick Check
2 5
18x y = 2 × 3 × 3 × x × x × y × y × y × y × y Factorise:
27x y = 3 × 3 × 3 × x × x × x × x × x × y × y 14xy – 21yz
5 2
2 2
HCF = 3 × 3 × x × x × y × y = 9x y
2 2
2
3
3
2 5
3 4
5 2
Therefore, 9x y + 18x y + 27x y = 9x y (xy + 2y + 3x ), the required factorise form.
When Each Term of the Given Expression Contains a Common Binomial Factor
To factorise this type of algebraic expression, follow these steps:
Step 1: Find the greatest binomial factor (i.e., the HCF) of all the terms.
Step 2: Take out the common factor and use the distributive property.

2
Example 2: Factorise: (a) 7a(3x + 4y) – b(3x + 4y) (b) (p – 2q) – 4p + 8q
Solution: (a) 7a(3x + 4y) – b(3x + 4y) = (3x + 4y)(7a – b)

2
2
(b) (p – 2q) – 4p + 8q = (p – 2q) – 4(p – 2q) = (p – 2q) {(p – 2q) – 4}
= (p – 2q) (p – 2q – 4), the required factorise form.
Factorisation by Regrouping the Terms

Sometimes algebraic expressions cannot be factorised by taking out their common factors. In
such cases, we will follow the following steps:

Step 1: Arrange the terms of the given algebraic expression in groups of terms such that all the
groups have a common factor.

Step 2: Find out the common factor from each group.
Step 3: Factorise each groups.

Example 3: Factorise each of the following algebraic expressions:

2
2
(a) 3a + ab + 3b + b (b) xy – yz – xz + xy + z – yz
2
2
Solution: (a) 3a + ab + 3b + b = 3a + 3b + ab + b = 3(a + b) + b (a + b) = (a + b) (3 + b)
(b) xy – yz – xz + xy + z – yz = xy + xy – yz – yz – xz + z = 2xy – 2yz – xz + z
2
2
2
= 2y(x – z) – z(x – z) = (x – z)(2y – z)
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