1MA1 · A4AlgebraFoundation & Higher

Expanding Brackets

Expanding replaces a product containing brackets with an equivalent sum of terms. Accuracy depends on multiplying every required pair and then collecting only genuinely like terms.

50 min Video by Maths Genie Pearson Edexcel GCSE Maths (1MA1)

The big picture

Expanding is the distributive law in action: a factor outside a bracket scales every term inside it. The same idea extends from a single bracket to two or even three binomials.

Edexcel assesses more than a mechanical grid. Expansions are used to prove identities, form equations from shapes and expose the structure of quadratic expressions, so clear algebraic working matters.

Expanding BracketsWatch the walkthrough, then read the notes below.

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What you'll learn

Expand a single bracket, including negative and algebraic factors
Expand and simplify expressions containing several brackets
Expand two binomials accurately
Extend the method to three binomials at Higher tier

Single brackets: distribute to every term

Multiply the factor outside by every term inside. Keep the sign attached to each term: $4 (2 x - 3) = 8 x - 12$ .

An algebraic factor changes coefficients and powers. For example, $3 x (2 x^{2} - 5 x + 4) = 6 x^{3} - 15 x^{2} + 12 x$ .

a (b + c) = ab + a c

The distributive law.

- 3 y \times 2 y^{2} = - 6 y^{3}

- 3 y \times (- 4 y) = + 12 y^{2}

- 3 y \times 5 = - 15 y

Answer

- 6 y^{3} + 12 y^{2} - 15 y

Why this works

The bracket groups a sum as one object. Multiplication must act on the whole group, so no term inside can be skipped.

Several single brackets

Expand each bracket separately, then collect like terms. A minus sign immediately before a bracket is equivalent to multiplying every term by $- 1$ .

1Expand the first bracket:

6 x - 15

2Expand the subtracted bracket:

- 2 x - 8

3Collect like terms:

6 x - 2 x - 15 - 8

Answer

4 x - 23

Tip — Write the sign produced by each multiplication before collecting terms. Mental sign changes are easy to lose.

Two binomials

Every term in the first bracket multiplies every term in the second. A grid or four connecting arcs makes the four products visible. Then collect the two middle like terms.

For $(x + a) (x + b)$ , the constant term is $ab$ and the coefficient of $x$ is $a + b$ . This provides a quick check, but the full method also works when the coefficient of $x^{2}$ is not 1.

(x + a) (x + b) = x^{2} + (a + b) x + ab

A useful monic quadratic pattern.

(a + b) (c + d) = a c + a d + b c + b d

All four products must appear.

2 x \times x = 2 x^{2}

and

2 x \times 5 = 10 x

- 3 \times x = - 3 x

and

- 3 \times 5 = - 15

3Collect the middle terms:

10 x - 3 x = 7 x

Answer

2 x^{2} + 7 x - 15

Why this works

The four products come directly from applying the distributive law twice. A grid is not a different rule; it is a way to organise that structure.

Three binomials (Higher)

Expand two brackets first and simplify the resulting quadratic. Then multiply every term of that quadratic by every term in the third bracket. Keep the work staged to avoid missing products.

1First,

(x + 1) (x + 2) = x^{2} + 3 x + 2

2Multiply by

(x - 3)

x (x^{2} + 3 x + 2) - 3 (x^{2} + 3 x + 2)

3Expand and collect:

x^{3} + 3 x^{2} + 2 x - 3 x^{2} - 9 x - 6

Answer

x^{3} - 7 x - 6

Think like an examiner

Show the unsimplified products before collecting like terms; this makes your method clear and helps recover marks from one sign error.
Only like terms can be collected: $x^{2}$ and $x$ have different powers and must remain separate.
When expanding three brackets, fully simplify the first pair before introducing the third.

Remember these

a (b + c) = ab + a c

Single-bracket expansion.

(x + a) (x + b) = x^{2} + (a + b) x + ab

Two monic binomials.

Watch out for these

3 (x + 4) = 3 x + 4

The 3 multiplies both terms, giving

3 x + 12

- 2 (x - 5) = - 2 x - 10

A negative times a negative is positive, so the answer is

- 2 x + 10

(x + 2) (x + 3) = x^{2} + 6

Include the two cross-products:

x^{2} + 3 x + 2 x + 6 = x^{2} + 5 x + 6

Stretch yourself

The expression $(x + a) (2 x - 3)$ expands to $2 x^{2} + 7 x - 15$ . Find $a$ and justify your answer.

Hint — Compare the constant term first, then verify the coefficient of $x$ .

Questions students ask

Do I have to use FOIL?

No. FOIL is a memory aid for two binomials. A grid or repeated distributive law is equally valid and generalises more reliably.

How can I check an expansion?

Substitute a simple value such as

x = 1

into both the original and expanded expressions. Equal results do not prove the expansion, but unequal results reveal an error.

Key takeaways

Multiply the outside factor by every term in a single bracket.
For two brackets, form every pairwise product and then collect like terms.
Treat negative signs as factors and keep multi-bracket expansions in clear stages.

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