YEAR 10 · ALGEBRA · 15 MIN

Expanding products

Use the distributive law to expand products accurately, including negative signs and variable factors.

In this lesson

Multiply every bracket term by the outside factor
Expand products and simplify without losing a sign

Before you start

Like terms
Signed multiplication

01 / LEARN

Understand the idea.

Distribute to every term

Distributing means multiplying the outside factor by every term inside the bracket. Treat subtraction as addition of a negative term: in $a (b - c)$ , the second product is $a (- c)$ . This keeps the signs visible before you simplify.

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

More than one bracket

When both factors contain sums, each term in the first bracket multiplies each term in the second. Two terms multiplied by two terms produce four products before any collection. A grid or organised list helps ensure every product appears once.

Simplify individual products first, including their signs and variable powers, then collect like terms. Expanding removes brackets but preserves the expression for every input; it does not introduce a new equation to solve.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

A variable multiplier

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

Worked solution

1
Distribute the whole outside factor.
$3 x (2 x - 5) = (3 x) (2 x) + (3 x) (- 5)$
Both terms in the bracket are multiplied by $3 x$ .
2
Simplify the two products.
$\begin{aligned} (3 x) (2 x) & = 6 x^{2} \\ (3 x) (- 5) & = - 15 x \end{aligned}$
Multiply coefficients and use $x \times x = x^{2}$ in the first product.
3
Write the expanded expression.
$6 x^{2} - 15 x$
The square and linear terms are unlike terms.

Answer

$6 x^{2} - 15 x$

EXAMPLE 02

A negative multiplier

Expand and simplify $- 2 (3 a - 4 b) + 5 a$ .

Worked solution

1
Multiply both bracket terms by negative two.
$- 2 (3 a - 4 b) + 5 a = (- 2) (3 a) + (- 2) (- 4 b) + 5 a$
The multiplier includes its negative sign.
2
Simplify the products.
$- 6 a + 8 b + 5 a$
A negative multiplied by a negative gives the positive $8 b$ term.
3
Collect the terms in a.
$(- 6 + 5) a + 8 b = - a + 8 b$
The extra $5 a$ combines with $- 6 a$ , while the $b$ term stays separate.

Answer

$- a + 8 b$

03 / YOUR TURN

Try it yourself.

Check your understanding

Expand $- 4 p (2 p - 3)$ .

Show answer & reasoning

$- 8 p^{2} + 12 p$

The two products are $(- 4 p) (2 p) = - 8 p^{2}$ and $(- 4 p) (- 3) = 12 p$ . Adding them gives $- 8 p^{2} + 12 p$ .

Keep practising

6 worksheet questions · 14 marks

Open worksheet Worked solutions

Read the questions online

Expand $5 (x + 3)$ .
2 marks
Expand $- 3 (2 a - 7)$ .
2 marks
Expand $2 x (4 x + 5)$ .
2 marks
Expand and simplify $4 (2 y - 1) - 3 (y + 2)$ .
2 marks
A shop packs $n$ identical trays. Each tray has $3 a + 2$ items. Another $4 n$ loose items are added. Expand and simplify the total.
3 marks
Expand $(2 a + b) (c - 3)$ without collecting unlike terms.
3 marks

Read the worked solutions

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

Expand $5 (x + 3)$ .
2 marks
Multiply both terms by $5$ .
$5 x + 15$
Expand $- 3 (2 a - 7)$ .
2 marks
Compute $(- 3) (2 a)$ and $(- 3) (- 7)$ .
$- 6 a + 21$
Expand $2 x (4 x + 5)$ .
2 marks
Multiply the coefficients and variable factors.
$8 x^{2} + 10 x$
Expand and simplify $4 (2 y - 1) - 3 (y + 2)$ .
2 marks
$8 y - 4 - 3 y - 6$ .
Collect the terms.
$5 y - 10$
A shop packs $n$ identical trays. Each tray has $3 a + 2$ items. Another $4 n$ loose items are added. Expand and simplify the total.
3 marks
$n (3 a + 2) + 4 n = 3 a n + 2 n + 4 n$ .
Combine the $n$ terms.
$3 a n + 6 n$
Expand $(2 a + b) (c - 3)$ without collecting unlike terms.
3 marks
The four products are $2 a c, - 6 a, b c, - 3 b$ .
$2 a c - 6 a + b c - 3 b$

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Curriculum reference & lesson scope

Australian Curriculum v9 and South Australian Mathematics Curriculum

Year 10 algebra, including the approved fuller school-course treatment of rational expressions and formula restrictions; quadratic methods are in their own chapter.

Official curriculum source ↗

In this lesson

Understand the idea.

Distribute to every term

More than one bracket

See the method in action.

A variable multiplier

Distribute the whole outside factor.

Simplify the two products.

Write the expanded expression.

A negative multiplier

Multiply both bracket terms by negative two.

Simplify the products.

Collect the terms in a.

Try it yourself.

Check your understanding

Keep practising