What does expand mean in maths?

Removing the brackets by multiplying everything inside by what's outside. Expanding 3(x + 4) gives 3x + 12. It's the opposite of factorising.

What is the FOIL method?

A way to remember the four multiplications when expanding two brackets: First, Outer, Inner, Last. For (x + 3)(x + 5): x × x, x × 5, 3 × x and 3 × 5, which gives x² + 8x + 15 once you collect like terms.

How do you expand (x + a)²?

Write it as (x + a)(x + a) and expand, or use the rule (x + a)² = x² + 2ax + a². So (x + 5)² = x² + 10x + 25. It is not x² + 25: that misses the middle term.

How do you expand a bracket with a negative in front?

Multiply every term inside by the negative number, so each sign flips. −2(x − 5) = −2x + 10, because −2 × −5 = +10.

What year level learns expanding brackets?

In the Victorian Curriculum 2.0, expanding with the distributive law is Year 8, expanding binomial products like (x + 3)(x + 5) is Year 9, and Year 10 uses perfect squares and difference of squares expansions with quadratics. Schools sequence topics differently, so check your course.

How to Expand Brackets: Distributive Law, FOIL and Worked Examples (Years 8 to 10)

Expanding a single bracket (the distributive law)

a(b + c) = ab + ac

Multiply the term outside by every term inside the bracket.

Worked examples

3(x + 4) = 3x + 12

5(2a - 3) = 10a - 15

x(x + 7) = x^2 + 7x

2a(3a - 4b) = 6a^2 - 8ab

When a letter is outside, use index laws: $x \times x = x^2$ and $2a \times 3a = 6a^2$ .

Expanding with a negative outside

The negative sign belongs to the number outside, and it multiplies every term inside. A negative times a negative is positive, so every sign inside flips.

Worked examples

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

-4(3x + 2) = -12x - 8

-(x - 6) = -x + 6

Expanding and simplifying

Expand each bracket, then collect like terms (the $x$ terms together and the numbers together).

Worked example

Expand and simplify $3(x + 2) + 4(x - 1)$ .

= 3x + 6 + 4x - 4

= 7x + 2

Worked example with a subtraction

Expand and simplify $5(2x - 1) - 3(x - 4)$ .

The $-3$ multiplies both terms in the second bracket, so $-3 \times -4 = +12$ .

= 10x - 5 - 3x + 12

= 7x + 7

Expanding two brackets (binomial products and FOIL)

(a + b)(c + d) = ac + ad + bc + bd

Every term in the first bracket multiplies every term in the second, so there are four multiplications. FOIL is a way to remember them: First, Outer, Inner, Last.

Worked example

Expand $(x + 3)(x + 5)$ .

First: $x \times x = x^2$ . Outer: $x \times 5 = 5x$ . Inner: $3 \times x = 3x$ . Last: $3 \times 5 = 15$ .

(x + 3)(x + 5) = x^2 + 5x + 3x + 15

= x^2 + 8x + 15

Worked examples with negatives

$(x - 6)(x - 2)$ : the last term is $-6 \times -2 = +12$ .

= x^2 - 2x - 6x + 12 = x^2 - 8x + 12

$(2x - 1)(x + 4)$ :

= 2x^2 + 8x - x - 4 = 2x^2 + 7x - 4

The area (grid) method

If FOIL feels easy to mix up, draw a grid. Put the terms of one bracket across the top and the other down the side, multiply to fill each box, then add everything up. Here's $(2x - 1)(x + 4)$ :

×	$x$	$+4$
$2x$	$2x^2$	$8x$
$-1$	$-x$	$-4$

Add the four boxes: $2x^2 + 8x - x - 4 = 2x^2 + 7x - 4$ .

The grid works for any size of bracket, so it's handy for three terms too.

Perfect squares and difference of squares

Some products come up so often they're worth knowing as rules.

Perfect squares

(a + b)^2 = a^2 + 2ab + b^2 \qquad (a - b)^2 = a^2 - 2ab + b^2

Worked examples

$(x + 5)^2 = (x + 5)(x + 5)$

= x^2 + 10x + 25

$(3x - 2)^2$ : square the first, double the product, square the last.

= 9x^2 - 12x + 4

Difference of squares

(a + b)(a - b) = a^2 - b^2

Worked examples

$(x + 7)(x - 7)$ : the middle terms $-7x$ and $+7x$ cancel.

= x^2 - 49

(2x + 3)(2x - 3) = 4x^2 - 9

Expanding three terms or three brackets

The rule doesn't change: every term in one bracket multiplies every term in the other. A binomial times a trinomial gives $2 \times 3 = 6$ products before you simplify.

Worked example

Expand $(x + 2)(x^2 + 3x + 1)$ .

Multiply by $x$ , then by $2$ :

= x^3 + 3x^2 + x + 2x^2 + 6x + 2

= x^3 + 5x^2 + 7x + 2

Worked example: three brackets

Expand $(x + 1)(x + 2)(x + 3)$ .

Expand two brackets first: $(x + 1)(x + 2) = x^2 + 3x + 2$ .

(x^2 + 3x + 2)(x + 3) = x^3 + 3x^2 + 3x^2 + 9x + 2x + 6

= x^3 + 6x^2 + 11x + 6

Common mistakes

Only multiplying the first term. $3(x + 4)$ is $3x + 12$ , not $3x + 4$ .
Losing the sign. In $5 - 2(x - 3)$ , the $-2$ multiplies the $-3$ to give $+6$ : $5 - 2x + 6 = 11 - 2x$ .
Squaring each term. $(x + 5)^2$ is not $x^2 + 25$ . Write it as two brackets and you'll see the $10x$ .
Forgetting to collect like terms at the end.

To check an expansion, substitute a number. With $x = 2$ : $(2 + 3)(2 + 5) = 35$ and $2^2 + 8 \times 2 + 15 = 35$ . They match.

Practice questions with answers

Try each one before you open the answer.

Expand $4(3x - 2)$ .
Show the answer
$12x - 8$
Expand $-3(2x - 5)$ .
Show the answer
$-6x + 15$
Expand and simplify $2(x + 5) - 3(x - 2)$ .
Show the answer
$2x + 10 - 3x + 6 = -x + 16$
Expand $(x + 4)(x + 6)$ .
Show the answer
$x^2 + 6x + 4x + 24 = x^2 + 10x + 24$
Expand $(3x - 2)(x + 5)$ .
Show the answer
$3x^2 + 15x - 2x - 10 = 3x^2 + 13x - 10$
Expand $(x - 7)^2$ .
Show the answer
$x^2 - 14x + 49$
Expand $(5x + 1)(5x - 1)$ .
Show the answer
Difference of squares: $25x^2 - 1$
Expand and simplify $(x + 3)(x^2 - 2x + 4)$ .
Show the answer
$x^3 - 2x^2 + 4x + 3x^2 - 6x + 12 = x^3 + x^2 - 2x + 12$

Expanding brackets by year level

Following the Victorian Curriculum 2.0. Schools sequence topics differently, so check your course.

Year 8

Algebra

The distributive law: expanding single brackets and collecting like terms.

Worksheets and topic test · Free PDF with answers

Year 9

Algebra

Expanding binomial products like (x + 3)(x + 5).

Worksheets and topic test · Free PDF with answers

Year 10

Quadratic factorising

Perfect squares and difference of squares, expanding and factorising quadratics.

Worksheets and topic test · Free PDF with answers

How to expand brackets

Expanding a single bracket (the distributive law)

Expanding with a negative outside

Expanding and simplifying

Expanding two brackets (binomial products and FOIL)

The area (grid) method

Perfect squares and difference of squares

Expanding three terms or three brackets

Common mistakes

Practice questions with answers

Expanding brackets by year level

Expanding brackets: common questions

More maths help, Years 7 to 10

Got an expanding question wrong in homework?