Factorizing an algebraic expression means finding all the factors of the expression and re writing it as a product of its factors.

There are many methods of factorizing an algebraic expression. In this topic, we use common terms, regrouping and identities to factorize an algebraic expression.

Factorization by using common terms

We know that 6 = 2 × 3. 2 and 3 are the factors of 6. So the term 6 can be factorized and re written as 6 = 2 × 3

Similarly, 6x if factorized as 6x = 2(3x) or 6x = 3(2x)

Example

Factorize

(a) 6x + 9

(b) −7a² + 21a

Solution

(a) There are two terms 6x and 9. Find the Highest Common Factor (HCF) that divides both terms of 6x and 9. The common factor is 3.

Write 3 outside a bracket as 3(? + ? )

Find the missing terms inside the bracket by dividing each term by the common factor.

That means 6x ÷ 3 = 2x and 9 ÷ 3 = 3. So 2x and 3 are the missing terms. Write these inside the bracket as 3(2x + 3)

Check your answer by expanding the bracket and see that the expanded form matches the original equation: 3(2x + 3) = (3 × 2x) + (3 × 3) = 6x + 9

(b) Here, 7 is a common factor of 7 and 21 and a is a common factor of a² and a.
Hence, −7a² + 21a = 7a (3 − a).