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Factorisation Using Identities

Factorisation using identities means rewriting an expression as a product by matching it with a known algebraic identity.

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Student-friendly explanation

Many expressions look difficult until the pattern is recognised. For example, x^2+10x+25 matches a^2+2ab+b^2, so it becomes (x+5)^2. The skill is to compare terms carefully before selecting the identity.

How to write this in exams

  1. 1

    Start with the exact idea

    Factorisation using identities means rewriting an expression as a product by matching it with a known algebraic identity.

  2. 2

    Then show how to use it

    Look for squares or cubes, test the middle term if present, choose the identity, write the product form, and verify by quick expansion.

  3. 3

    Add one concrete example

    9a^2-25 = (3a)^2-5^2 = (3a-5)(3a+5).

  4. 4

    Avoid this incomplete answer

    Writing x^2+8x+16 as (x+8)^2 is wrong because (x+8)^2 gives x^2+16x+64.

Definition

Factorisation using identities means rewriting an expression as a product by matching it with a known algebraic identity.

Example

9a^2-25 = (3a)^2-5^2 = (3a-5)(3a+5).

Rule to remember

Match a^2+2ab+b^2, a^2-2ab+b^2, a^2-b^2, a^3+b^3, or a^3-b^3 before factorising.

Memory hook

Pattern first, identity next, factor last.

Examples and method

Worked example

Factorise 4x^2+20x+25. The first and last terms are (2x)^2 and 5^2, and the middle term is 2(2x)(5)=20x. So the factor form is (2x+5)^2.

Method to apply

Look for squares or cubes, test the middle term if present, choose the identity, write the product form, and verify by quick expansion.

How CBSE asks it

Students may get direct factorisation, simplification before substitution, or cancellation in rational expressions.

Avoid common mistakes

Common confusion

Students often choose an identity only by seeing two square terms and ignore the middle term condition.

Common wrong answer

Writing x^2+8x+16 as (x+8)^2 is wrong because (x+8)^2 gives x^2+16x+64.

Exam tip

Before factorising a three-term expression as a perfect square, check whether the middle term is exactly twice the product of the square roots.

Quick check

How will you factorise x^2-12x+36 using an identity?

x^2-12x+36 is x^2-2(x)(6)+6^2, so it matches (a-b)^2 and factorises as (x-6)^2.

Answer writing and exam use

1-mark answer

Factorisation using identities means rewriting an expression as a product by matching it with a known algebraic identity.

2-mark answer

Factorisation using identities means rewriting an expression as a product by matching it with a known algebraic identity. Match a^2+2ab+b^2, a^2-2ab+b^2, a^2-b^2, a^3+b^3, or a^3-b^3 before factorising. 9a^2-25 = (3a)^2-5^2 = (3a-5)(3a+5).

3-mark answer

Many expressions look difficult until the pattern is recognised. For example, x^2+10x+25 matches a^2+2ab+b^2, so it becomes (x+5)^2. The skill is to compare terms carefully before selecting the identity. Match a^2+2ab+b^2, a^2-2ab+b^2, a^2-b^2, a^3+b^3, or a^3-b^3 before factorising. Factorise 4x^2+20x+25. The first and last terms are (2x)^2 and 5^2, and the middle term is 2(2x)(5)=20x. So the factor form is (2x+5)^2. Students may get direct factorisation, simplification before substitution, or cancellation in rational expressions. Writing x^2+8x+16 as (x+8)^2 is wrong because (x+8)^2 gives x^2+16x+64.
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