YEAR 8 · ALGEBRAIC EXPRESSIONS · 13 MIN

Expanding a single bracket

Expand a single bracket by multiplying every term, including signed terms, by the outside factor.

In this lesson

Apply the distributive property
Expand negative factors
Collect terms after expansion

Before you start

Algebraic multiplication
Signed multiplication

01 / LEARN

Understand the idea.

The factor reaches every term

Expanding uses the distributive property $a (b + c) = a b + a c$ . The outside factor multiplies every term inside the bracket. Thus $4 (x + 3)$ becomes $4 x + 12$ : both x and the constant three receive the factor four.

Expansion changes a product into an equivalent sum. It does not find a numerical value for the variable. The original and expanded forms agree for every allowed input because each term has been multiplied according to the distributive property.

Distribute signs, then collect

Treat subtraction inside a bracket as a signed term. In $- 2 (3 x - 5)$ , multiply $- 2$ by $3 x$ and by $- 5$ . The second product is positive because both of its factors are negative.

After expanding, collect like terms with any terms already outside the bracket. Keep their original signs. Substitution can help spot an error, but the distributive steps themselves explain why the two expressions are equal for every input.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Expand then simplify

Simplify $3 (x + 4) + 2 x$ .

Worked solution

1
Distribute three across both bracket terms.
$3 (x + 4) + 2 x = 3 x + 3 (4) + 2 x$
Both x and four belong to the bracket multiplied by three.
2
Complete the constant product.
$3 x + 3 (4) + 2 x = 3 x + 12 + 2 x$
The added two-x term stays outside the expansion and is unchanged.
3
Collect like terms.
$3 x + 12 + 2 x = (3 + 2) x + 12 = 5 x + 12$
Only the two x terms combine; the constant remains separate.

Answer

$5 x + 12$

EXAMPLE 02

Negative outside factor

Expand $- 3 (2 y - 5)$ .

Worked solution

1
Multiply the first term by negative three.
$(- 3) (2 y) = - 6 y$
The positive variable term receives a negative coefficient.
2
Multiply the signed constant by negative three.
$(- 3) (- 5) = 15$
A negative factor times a negative factor gives a positive result.
3
Add the two products.
$- 3 (2 y - 5) = - 6 y + 15$
The distributive property includes one product for each term in the bracket.

Answer

$- 6 y + 15$

03 / YOUR TURN

Try it yourself.

Check your understanding

Expand $- 2 (x - 6)$ .

Show answer & reasoning

$- 2 x + 12$

Multiply negative two by both signed terms: $(- 2) x = - 2 x$ and $(- 2) (- 6) = 12$ . The expansion is $- 2 x + 12$ .

Keep practising

6 worksheet questions · 13 marks

Open worksheet Worked solutions

Read the questions online

Expand $4 (x + 2)$ .
2 marks
Expand $3 (2 a - 5)$ .
2 marks
Expand $- 5 (y + 3)$ .
2 marks
Expand $- 2 (4 p - 7)$ .
2 marks
Simplify $2 (x + 3) + 5 x$ .
2 marks
A strip is $3$ cm wide and $(n + 4)$ cm long. Write its area in expanded form.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Expand $4 (x + 2)$ .
2 marks
Multiply both terms by $4$ .
$4 x + 8$
Expand $3 (2 a - 5)$ .
2 marks
$3 (2 a) = 6 a$ and $3 (- 5) = - 15$ .
$6 a - 15$
Expand $- 5 (y + 3)$ .
2 marks
Distribute $- 5$ to both terms.
$- 5 y - 15$
Expand $- 2 (4 p - 7)$ .
2 marks
$(- 2) (4 p) = - 8 p$ and $(- 2) (- 7) = 14$ .
$- 8 p + 14$
Simplify $2 (x + 3) + 5 x$ .
2 marks
Expand to $2 x + 6 + 5 x$ .
Collect the $x$ terms.
$7 x + 6$
A strip is $3$ cm wide and $(n + 4)$ cm long. Write its area in expanded form.
3 marks
Area is $3 (n + 4)$ .
Distribute to get $3 n + 12$ .
$(3 n + 12)$ square centimetres

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Curriculum reference & lesson scope

Australian Curriculum v9 Mathematics with SA curriculum context

Combining like terms; substitution with signed/rational values; multiplying/dividing algebraic terms; expanding a single bracket; factorising common factors; equivalent linear expressions; rearranging simple formulas.

Official curriculum source ↗

In this lesson

Understand the idea.

The factor reaches every term

Distribute signs, then collect

See the method in action.

Expand then simplify

Distribute three across both bracket terms.

Complete the constant product.

Collect like terms.

Negative outside factor

Multiply the first term by negative three.

Multiply the signed constant by negative three.

Add the two products.

Try it yourself.

Check your understanding

Keep practising