ICSE Class 9 Maths Algebra: Easy Expansions & Factorisation

Ever wondered how to turn (x+3)(x-2) into a neat polynomial in seconds? Or how to break a messy expression back into simple pieces? Let’s make expansions and factorisation feel like a fun puzzle.

In simple words, expansion means opening up brackets to write a product as a single sum. Factorisation is the reverse – spotting a common piece and pulling it out, just like un‑wrapping a gift to see the present inside.

What is Expansion in Algebra?

Expansion is the process of removing brackets (parentheses) by multiplying each term inside them. Think of it like spreading butter on toast – you take the butter (the brackets) and spread it evenly over the whole slice (the expression).

Common expansion formulas you should remember

Notice the pattern? The middle terms have coefficients 2 or 3, matching the power of the binomial. Memorising these five formulas saves you loads of time during the exam.

How to Factorise an Expression?

Factorisation is the art of writing an expression as a product of simpler factors. Imagine you have a Lego model (the expression) and you want to separate it into its original bricks (the factors). The key steps are spotting a common factor – a number or variable that appears in every term – and then applying special product formulas.

Step‑by‑step factorisation flowchart

Follow the chart each time you meet a new expression. Let’s walk through a couple of examples.

Worked example 1: Taking out the common factor

Factorise 6x3 + 9x2 − 12x.

1. Identify the greatest common factor (GCF). All terms share a 3 and an x, so GCF = 3x.
2. Divide each term by 3x:
   6x3 ÷ 3x = 2x2;
   9x2 ÷ 3x = 3x;
   ‑12x ÷ 3x = ‑4.
3. Write the expression as 3x(2x2 + 3x ‑ 4).
4. Check if the quadratic inside can be factorised further. It doesn’t fit any special product, so we stop.

Result: 3x(2x2 + 3x ‑ 4).

Worked example 2: Using a special product

Factorise x2 ‑ 25.

• This matches the difference of squares formula: a2 ‑ b2 = (a ‑ b)(a + b).
• Here a = x and b = 5, so we get (x‑5)(x+5).

Result: (x‑5)(x+5).

Popular factorisation tricks

• Group and factor: For four‑term expressions, group the first two and the last two, factor each group, then look for a common binomial.
• Quadratic in disguise: If you see something like 4y4 ‑ 9, treat y2 as a single variable (let t = y2) and apply the difference of squares.
• Sum/Difference of cubes: a3 ± b3 = (a ± b)(a2 ∓ ab + b2). Remember the signs flip in the middle term.

Quick comparison: Expansion vs. Factorisation

Remember: expansion spreads, factorisation gathers. Switch between them as the problem demands.

💡 Tips to ace the ICSE exam

• Memorise the five binomial formulas; they appear in almost every paper.
• Always write the GCF first – it saves you from missing a hidden factor.
• After factorising, multiply the factors back (quick check) to avoid careless mistakes.
• Practice both forward (expand) and backward (factorise) directions; the two skills reinforce each other.

📝 Likely Exam Questions

1. Expand and simplify: (2x + 3)² ‑ (2x ‑ 3)².
   Answer: Use (a+b)² ‑ (a‑b)² = 4ab. Here a = 2x, b = 3, so result = 4·2x·3 = 24x.
2. Factorise completely: 4y⁴ ‑ 9.
   Answer: Treat y² as t. 4t² ‑ 9 = (2t ‑ 3)(2t + 3) → replace t = y² → (2y² ‑ 3)(2y² + 3).
3. Write the product form of (x + 5)(x ‑ 5)(x + 2).
   Answer: First two give x² ‑ 25. Multiply by (x + 2): (x² ‑ 25)(x + 2) = x³ + 2x² ‑ 25x ‑ 50.
4. Factorise: 6a³b ‑ 9a²b² + 3ab³.
   Answer: GCF = 3ab. Inside: 2a² ‑ 3ab + b² = (2a ‑ b)(a ‑ b). Final: 3ab(2a ‑ b)(a ‑ b).
5. Expand: (x ‑ 1)³.
   Answer: Use (a‑b)³ = a³ ‑ 3a²b + 3ab² ‑ b³ → x³ ‑ 3x² + 3x ‑ 1.