Sec 2 Expansion and Factorisation | Algebraic Products, Common Factors and Quadratic Foundations

Secondary 2 Expansion and Factorisation is the algebraic bridge between simple expressions and more powerful equation solving. Expansion takes a product and rewrites it as a sum. Factorisation reverses that process: it identifies the common or structured product hidden inside an expression.

Students searching for Sec 2 factorisation, expansion, algebraic products, common factor or factorising expressions often learn the two skills as separate procedures. The stronger understanding is that they are inverse representations of the same algebraic structure.

This page is the canonical Expansion and Factorisation owner under the Secondary Mathematics Topic Library. It depends on Sec 1 Algebraic Expressions and prepares students for equations, quadratics and later A-Math.

Quick answer: expansion and factorisation are opposites

Expansion: 3(x+4) = 3x+12.

Factorisation: 3x+12 = 3(x+4).

One distributes multiplication; the other extracts a common factor.

The distributive law

a(b+c) = ab+ac.

a(b−c) = ab−ac.

The multiplier outside the bracket applies to every term inside.

Worked expansion example

Expand: 5(2x−3).

5×2x = 10x.

5×(−3) = −15.

Answer: 10x−15.

Negative multiplier

Expand: −3(x−5).

−3x +15.

The second term becomes positive because negative × negative = positive.

Two brackets

For simple binomial products, every term in the first bracket multiplies every term in the second.

Example: (x+2)(x+5).

x·x +5x +2x +10 = x²+7x+10.

A grid method

A multiplication grid is a reliable way to expand two brackets. Put the terms of one bracket across the top and the other down the side, multiply each cell, then combine like terms.

This reduces missed products.

Common-factor factorisation

Factorise: 12x+18.

HCF of 12 and 18 is 6.

12x+18 = 6(2x+3).

The common factor is the largest factor shared by every term.

Common variable factors

Factorise: 8x²+12x.

Both terms contain 4x.

Answer: 4x(2x+3).

Factorisation as reverse expansion

Always check by expanding the factorised form.

4x(2x+3) = 8x²+12x.

Reverse checking is one of the best algebra error controls.

Difference between simplifying and factorising

Simplifying combines equivalent algebraic structure. Factorising rewrites an expression as a product.

For example, 3x+2x simplifies to 5x. But 6x+9 factorises to 3(2x+3).

Why factorisation matters later

solving quadratic equations
simplifying algebraic fractions
identifying common structure
graph intercepts and roots
Additional Mathematics algebra

Factorising simple quadratics

Example: x²+7x+12.

Find two numbers that multiply to 12 and add to 7: 3 and 4.

So x²+7x+12 = (x+3)(x+4).

This idea leads directly into the Quadratic Equations and Graphs owner.

Common errors

multiplies only the first term inside a bracket
loses a negative sign
does not combine like terms after expansion
extracts a factor that is not common to every term
factorises but leaves a larger common factor inside
cannot verify by re-expansion

The expansion-factorisation audit

If expanding, did every term multiply every required term?
Are signs correct?
Have like terms been combined?
If factorising, what is the greatest common factor?
Does re-expansion reproduce the original expression?

Practice

Practice 1

Question: Expand 4(x+3)

Answer: 4x+12

Practice 2

Question: Expand −2(3x−5)

Answer: −6x+10

Practice 3

Question: Expand (x+3)(x+2)

Answer: x²+5x+6

Practice 4

Question: Factorise 10x+15

Answer: 5(2x+3)

Practice 5

Question: Factorise 6x²+9x

Answer: 3x(2x+3)

Practice 6

Question: Factorise x²+9x+20

Answer: (x+4)(x+5)

Frequently asked questions

What is the difference between expansion and factorisation?

They are reverse processes: expansion removes brackets; factorisation creates a product form.

How do I check factorisation?

Expand the answer and compare with the original expression.

Why find the HCF first?

It gives the largest common factor that can be extracted cleanly.

Why is factorisation important?

It becomes essential for quadratics, algebraic fractions and later advanced algebra.

Where does this sit in Atlas?

This is the canonical Sec 2 Expansion and Factorisation owner under Secondary Mathematics Topic Library.

The final Expansion rule

See expansion and factorisation as two views of the same structure. If one direction becomes uncertain, reverse the process to check the other.

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