Factorisation

• How to write an algebraic expression as a product of factors — the reverse of expansion.
• Methods: common factor, regrouping, and using identities in reverse.
• How factorisation simplifies fractions, solves equations, and reveals structure in patterns.
• CBSE techniques for binomials and trinomials in one variable.

1. Definition — Factorisation means expressing an expression as a product of simpler expressions (factors). Example: 2x + 6 = 2(x + 3).

2. Common factor method — Take out the HCF of all terms. 6x² + 9x = 3x(2x + 3).

3. Regrouping — Split terms into groups that share a common factor. ax + ay + bx + by = a(x + y) + b(x + y) = (a + b)(x + y).

4. Using identities (reverse) — Recognise perfect squares and difference of squares:

   • a² + 2ab + b² = (a + b)²
   • a² − 2ab + b² = (a − b)²
   • a² − b² = (a + b)(a − b)

5. Factorising trinomials — x² + 5x + 6 = (x + 2)(x + 3) by finding two numbers that multiply to 6 and add to 5.

6. Where it shows up — Simplifying algebraic fractions, finding dimensions when area is given as a polynomial, and solving quadratic equations in higher classes.

Factorise 4x² − 12x + 9 completely.

Step 1 — Check: first term 4x² = (2x)², last term 9 = 3²
Step 2 — Middle term: −12x = 2(2x)(−3)  → matches a² − 2ab + b²
Step 3 — Apply identity: (2x − 3)²
Answer: (2x − 3)²
Verify: (2x − 3)(2x − 3) = 4x² − 12x + 9 ✓

Application: A square garden has side (2x − 3) m. Its area is (2x − 3)² = 4x² − 12x + 9 m² — factorisation connects side length to area.

• Stopping before fully factored (6x + 12 = 2(3x + 6) should be 6(x + 2)).
• Wrong signs in (a − b)²: it expands to a² − 2ab + b², not a² + b².
• Forgetting to take out the largest common factor first.
• Treating x² + 4 as (x + 2)² — it is not a perfect square trinomial.

• Factorise 5x + 15.
• Factorise x² − 9 using an identity.
• Factorise x² + 7x + 12.
• Factorise 2ax + 2ay + bx + by by regrouping.

Open the Practice tab for graded questions on Factorisation.