Expanding and factorising

Q: What does the 'FOIL' acronym stand for when expanding double brackets?

FOIL stands for First , Outside , Inside , and Last , representing the order of terms to multiply.

Q: What is the 'Sum and Product Rule' for factorising quadratics of the form \(x^2 + bx + c\)?

You must find two numbers that multiply to give the constant term (\(c\)) and add to give the coefficient of the middle term (\(b\)).

Mathematics (1MA1) · Pearson Edexcel GCSE (9-1) · 5 min read · Updated: 7 Sept 2026

Introduction to Expanding and Factorising

Welcome to one of the most important chapters in GCSE Algebra! Think of expanding and factorising as two sides of the same coin. Expanding is like "unwrapping" a present to see what is inside, while factorising is like "re-packing" it into a neat box. These skills are the building blocks for solving equations and understanding graphs, which you will use throughout your 1MA1 course.

1. Expanding Single Brackets

Expanding a bracket means multiplying everything inside the bracket by the term on the outside. This is often called the distributive law.

How to do it:

1. Look at the term outside the bracket.
2. Multiply it by the first term inside.
3. Multiply it by the second term inside (watch out for the plus or minus sign!).

Example 1: Expand \(3(x + 5)\)
Multiply \(3 \times x = 3x\)
Multiply \(3 \times 5 = 15\)
Answer: \(3x + 15\)

Example 2: Expand \(x(x - 4)\)
Multiply \(x \times x = x^2\)
Multiply \(x \times -4 = -4x\)
Answer: \(x^2 - 4x\)

Quick Tip: Don't forget the signs! A negative times a negative makes a positive. For example, \(-2(x - 3)\) becomes \(-2x + 6\).

2. Factorising into Single Brackets

Factorising is the exact opposite of expanding. We look for the Highest Common Factor (HCF) of the terms and put it outside the bracket.

Step-by-Step:

1. Find the biggest number that divides into all the coefficients (the numbers).
2. Look for any letters (variables) that appear in every term.
3. Put these "common factors" outside the bracket.
4. Divide each original term by the factor outside to see what is left inside.

Example 1: Factorise \(10x - 15\)
The HCF of \(10\) and \(15\) is \(5\).
Divide \(10x\) by \(5\) to get \(2x\).
Divide \(-15\) by \(5\) to get \(-3\).
Answer: \(5(2x - 3)\)

Example 2: Factorise \(x^2 + 7x\)
Both terms have an \(x\), so \(x\) goes outside.
\(x^2 \div x = x\)
\(7x \div x = 7\)
Answer: \(x(x + 7)\)

Key Takeaway: You can always check your factorising by expanding your answer. If you don't get the original expression back, something went wrong!

3. Expanding Two Binomials (Double Brackets)

A binomial is just a fancy name for an expression with two terms, like \((x + 2)\). When we multiply two of these together, every term in the first bracket must meet every term in the second bracket.

The "FOIL" Method:

Many students find it helpful to remember the word FOIL:

First: Multiply the first terms in each bracket.
Outside: Multiply the two outermost terms.
Inside: Multiply the two innermost terms.
Last: Multiply the last terms in each bracket.

Example: Expand \((x + 3)(x + 5)\)
F: \(x \times x = x^2\)
O: \(x \times 5 = 5x\)
I: \(3 \times x = 3x\)
L: \(3 \times 5 = 15\)
Now, collect like terms: \(x^2 + 5x + 3x + 15 = x^2 + 8x + 15\).

Higher Tier Only: Expanding Three Brackets
If you have three brackets, like \((x+1)(x+2)(x+3)\), expand the first two first, simplify the result, and then multiply that new expression by the third bracket.

4. Factorising Quadratics (\(x^2 + bx + c\))

When we factorise a quadratic expression like \(x^2 + 7x + 10\), we want to put it back into two brackets: \((x + ?)(x + ?)\).

The Sum and Product Rule:

We need to find two numbers that:
1. Multiply to give the last number (\(c\)).
2. Add up to give the middle number (\(b\)).

Example: Factorise \(x^2 + 7x + 10\)
We need two numbers that multiply to \(10\) and add to \(7\).
Factors of \(10\) are: \(1, 10\) (sum \(11\)) and \(2, 5\) (sum \(7\)).
The numbers are \(2\) and \(5\).
Answer: \((x + 2)(x + 5)\)

Common Mistake: Be careful with negative signs! If the quadratic is \(x^2 - 5x + 6\), you need two numbers that multiply to \(+6\) but add to \(-5\). In this case, they are \(-2\) and \(-3\).

5. The Difference of Two Squares (DOTS)

This is a special pattern you must learn to recognise. It happens when you have a squared term minus another squared term, with no "middle" \(x\) term.

The rule is: \(a^2 - b^2 = (a + b)(a - b)\)

Example 1: Factorise \(x^2 - 16\)
Both \(x^2\) and \(16\) are square numbers (\(16 = 4^2\)).
Answer: \((x + 4)(x - 4)\)

Example 2: Factorise \(25y^2 - 81\)
\(\sqrt{25y^2} = 5y\) and \(\sqrt{81} = 9\).
Answer: \((5y + 9)(5y - 9)\)

6. Advanced Factorising (Higher Tier Only)

Higher tier students need to be able to factorise quadratics where the coefficient of \(x^2\) is greater than 1, such as \(2x^2 + 7x + 3\).

Method:

1. Multiply the first number (\(a\)) by the last number (\(c\)). For \(2x^2 + 7x + 3\), \(2 \times 3 = 6\).
2. Find two numbers that multiply to \(6\) and add to the middle number (\(7\)). These are \(6\) and \(1\).
3. Split the middle term: \(2x^2 + 6x + 1x + 3\).
4. Factorise in pairs: \(2x(x + 3) + 1(x + 3)\).
5. The final brackets are \((2x + 1)(x + 3)\).

Quick Review Checklist

- Single Bracket Expansion: Multiply everything inside by the term outside.
- Single Bracket Factorising: Take out the Highest Common Factor.
- Double Bracket Expansion: Use FOIL and simplify.
- Quadratic Factorising: Find two numbers that multiply to the end and add to the middle.
- Difference of Two Squares: Look for \(x^2\) minus a square number.
- Higher Tier: Practice expanding three brackets and factorising \(ax^2 + bx + c\).

Don't worry if this seems tricky at first! Algebra is a language; the more you "speak" it by practicing questions, the more natural it will feel.

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What is the 'distributive law' in the context of expanding single brackets?

The distributive law means multiplying the term on the outside of the bracket by every individual term inside the bracket.

What are the basic steps for factorising an expression into a single bracket?

Find the Highest Common Factor (HCF) of all terms, place it outside the bracket, and divide each term by this HCF to determine the terms inside.

What does the 'FOIL' acronym stand for when expanding double brackets?

FOIL stands for First, Outside, Inside, and Last, representing the order of terms to multiply.

How do you expand an expression containing three brackets, such as \((x+1)(x+2)(x+3)\)?

Expand the first two brackets first, simplify the resulting expression, and then multiply that simplified result by the third bracket.

What is the 'Sum and Product Rule' for factorising quadratics of the form \(x^2 + bx + c\)?

You must find two numbers that multiply to give the constant term (\(c\)) and add to give the coefficient of the middle term (\(b\)).

How do you factorise a 'Difference of Two Squares' (DOTS) expression?

The pattern is \(a^2 - b^2 = (a + b)(a - b)\). Find the square root of both terms and place them in brackets with alternating signs.

What is the method for factorising a quadratic where the \(x^2\) coefficient is greater than 1 (\(ax^2 + bx + c\))?

Multiply \(a\) and \(c\), find factors that add to \(b\), split the middle term, and factorise in pairs.

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