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Key points

Open bracket three n plus two close bracket open bracket n minus four close bracket. Three n plus two is in orange and n minus four is in blue. Underneath, a blank two by two grid with n on top of the first column and minus four on top of the second column – these are highlighted blue. Three n is to the left of the first row and two is to the left of the second row – these are highlighted orange. There is a multiplication symbol to the top left near the grid. — Image caption,
A grid can be used to expand double brackets. Every term in the first bracket must be multiplied by every term in the second bracket.

When expanding brackets it’s important to have a good understanding of multiplying terms, multiplying by positive and negative numbers and index notation.
To expand a single bracket, each inside the bracket is multiplied by the outside the bracket.
It may be possible to the expression after expanding the bracket. This is done by collecting the .
One bracket may be multiplied by another bracket. This is sometimes called expanding double brackets.
To expand double brackets, every term in the first bracket must be multiplied by every term in the second bracket. It may be possible to simplify the resulting expression.
The expression that results from expanding a single or a double bracket and the original expression are . They are expressions.

Open bracket three n plus two close bracket open bracket n minus four close bracket. Three n plus two is in orange and n minus four is in blue. Underneath, a blank two by two grid with n on top of the first column and minus four on top of the second column – these are highlighted blue. Three n is to the left of the first row and two is to the left of the second row – these are highlighted orange. There is a multiplication symbol to the top left near the grid. — Image caption,
A grid can be used to expand double brackets. Every term in the first bracket must be multiplied by every term in the second bracket.

Expanding a single bracket

To expand a single bracket multiply each term inside the bracket by the term outside the bracket.
If the term and the bracket are joined by an addition or subtraction symbol, the term is not multiplied by the expression in the bracket.
An addition sign before the bracket means each term is multiplied by 1. A subtraction sign in front of the bracket means each term is multiplied by –1.
When an expression includes multiple single brackets:

Expand each bracket in the expression by multiplying each term in the bracket by the term in front of the bracket.
Simplify the expression by collecting the like terms.

The starting expression, with a bracket (or brackets), is equivalent to the resulting expression. They are identities.

Examples

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Image caption,
Expand the expression 5(3𝒏 – 4).
Image caption,
The term in front of the bracket (5) is multiplied by each term inside the bracket. The calculations are 5 × 3𝒏 and 5 × –4
Image caption,
5 × 3𝒏 = 15𝒏. 5 × –4 = –20. 5(3𝒏 – 4) is equivalent to 15𝒏 –20. They are identities.
Image caption,
Expand 2𝒏(3𝒏 + 2).
Image caption,
The term in front of the bracket (2𝒏) is multiplied by each term inside the bracket. The calculations are 2𝒏 × 3𝒏 and 2𝒏 × 2
Image caption,
2𝒏 × 3𝒏 = 6𝒏². 2𝒏 × 2 = 4𝒏. 2𝒏(3𝒏 + 2) is equivalent to 6𝒏² + 4𝒏. They are identities.
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If there is an addition or subtraction symbol between a term and a bracket, the term is separated from the bracket. The bracket is not multiplied by that term. The expression can be written without the bracket. An addition sign before the bracket means each term is multiplied by 1. The terms stay the same, + 3𝒏 – 4. As this includes a subtraction the terms change signs. 3𝒏 becomes – 3𝒏 and – 4 becomes + 4. The expressions must now be simplified.
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The expressions are simplified by collecting the like terms. The sign in front of each term stays with that term. For the first expression, 5 – 4 = 1 and 5 + (3𝒏 – 4) becomes 3𝒏 + 1. For the second expression, 5 + 4 = 9 and 5 – (3𝒏 – 4) becomes 9 – 3𝒏. This may also be written as – 3𝒏 + 9
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Expand and simplify 3(𝒏 + 2) – 2(𝒏 – 3).
Image caption,
Expand each bracket in the expression. Multiply each term in the bracket by the term attached to the front of the bracket. For the first bracket, 3 is multiplied by 𝒏 and 2. For the second bracket, –2 is multiplied by 𝒏 and –3. The expression is now 3𝒏 + 6 – 2𝒏 + 6. This expression can be simplified by collecting the like terms. 3𝒏 – 2𝒏 = 𝒏 and 6 + 6 = 12. This expression can be simplified. 3(𝒏 + 2) – 2(𝒏 – 3) is equivalent to 𝒏 + 12. They are identities.

Questions

Question 1: Expand the bracket.

Two n open bracket eight n plus seven close bracket.

Question 2: Expand and simplify the expression.

Two plus three open bracket n minus seven close bracket minus n.

Expanding double brackets

To expand double brackets using a grid:

Multiply each term in the first bracket by each term in the second bracket. When there are two terms in each bracket, use a 2 x 2 grid. The dimensions of the grid match the number of terms in each bracket.
Write the terms for the first bracket on the left side of the grid and the terms in the second bracket across the top of the grid.
Fill in the grid by multiplying each term in the first bracket by each term in the second bracket.
Write the expression with all four terms.
Simplify the expression by collecting the like terms.

To expand double brackets without using a grid:

Multiply each term in the first bracket by each term in the second bracket.
Rewrite the expression so that the first bracket is split to show each term is multiplied by the second bracket. The expression now has two single brackets to expand.
Expand each single bracket.
Simplify the expression by collecting the like terms. The starting expression (with brackets) is equivalent to the resulting expression. They are identities.

Examples

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Image caption,
Expand (3𝒏 + 2)(𝒏 – 4).
Image caption,
Each term in the first bracket is multiplied by each term in the second bracket. There are two terms in each bracket so a 2 x 2 grid is used. Write the terms in the first bracket to the left of the grid and the terms in the second bracket across the top of the grid.
Image caption,
Complete the grid by multiplying each term in the first bracket by each term in the second bracket. 3𝒏 × 𝒏 = 3𝒏². 3𝒏 × –4 = –12𝒏. 2 × 𝒏 = 2𝒏. 2 × –4 = –8
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Write the expression with all four terms. Simplify the expression by collecting the like terms. (3𝒏 + 2)(𝒏 – 4) is equivalent to 3𝒏² – 10𝒏 – 8. They are identities.
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Expand and simplify (𝒏 + 6)(𝒏 – 4) without a grid.
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Rewrite the expression. The first term in the first bracket (𝒏) is multiplied by the second bracket, this is 𝒏(𝒏 – 4). The second term in the first bracket (6), is multiplied by the second bracket, this is + 6(𝒏 – 4). The expression now has two single brackets to expand.
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Expand each single bracket. Each term in (𝒏 – 4) is multiplied by 𝒏. Each term in (𝒏 – 4) is multiplied by 6
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(𝒏 + 6)( 𝒏 – 4) expands to give the expression 𝒏² – 4𝒏 + 6𝒏 – 24. This expression can be simplified.
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𝒏² – 4𝒏 + 6𝒏 – 24 can be simplified by collecting the like terms ( – 4𝒏 and 6𝒏). (𝒏 +6)(𝒏 – 4) is equivalent to 𝒏² + 2𝒏 – 24. They are identities.

Questions

Question 1: Expand and simplify the expression.

Open bracket p plus five closed bracket squared

Question 2: Expand and simplify the expression.

Open bracket n minus five close bracket open bracket n minus six close bracket.

Practise multiplying brackets

Quiz

Practise multiplying brackets with this quiz. You may need a pen and paper to help you work out your answers.

Game - Divided Islands

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