Expanding products | Year 10

01 / LEARN

The idea, step by step

Distribute to every term

In $a (b + c)$ , the outside factor multiplies both terms: $a b + a c$ . The same rule works with subtraction, negative coefficients and variable factors. Writing the two products before simplifying makes sign errors easier to catch.

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

More than one bracket

For two factors containing sums, multiply each term in the first bracket by each term in the second. Organise the products in a short list or grid, then collect like terms. Quadratic products and their factorisation are explored further in Quadratics and Parabolas; the distributive law is the common principle.

A common trap

$- 3 (x - 4) = - 3 x - 12$ .

Instead: The second product is $(- 3) (- 4) = + 12$ , so the result is $- 3 x + 12$ .

02 / WORKED EXAMPLES

See the thinking

Example 1

A variable multiplier

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

Multiply $3 x$ by $2 x$ to obtain $6 x^{2}$ .
Multiply $3 x$ by $- 5$ to obtain $- 15 x$ .

6 x^{2} - 15 x

Example 2

A negative multiplier

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

Distribute $- 2$ : $- 6 a + 8 b + 5 a$ .
Collect $- 6 a + 5 a = - a$ .

- a + 8 b

Check your understanding

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

Show answer

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

Both bracket terms are multiplied by $- 4 p$ ; the negative times negative gives $+ 12 p$ .

03 / WORKSHEET

Your turn

6 questions · 14 marks · Work at your own pace.

Worksheet PDF 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 ↗

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