Multiplying/dividing algebraic terms | Year 8

01 / LEARN

The idea, step by step

Multiply factors

An algebraic term is a product. For $3 a \times 4 b$ , the coefficients multiply to $12$ and the variable factors give $a b$ , so the result is $12 a b$ . A repeated variable becomes a power: $x \times x = x^{2}$ .

Multiplication differs from collection: $3 x + 4 x = 7 x$ , but $(3 x) (4 x) = 12 x^{2}$ . Use brackets when multiplying a term with a negative coefficient.

Divide by common factors

A fraction makes cancellation clear. In $\frac{15 x y}{5 x}$ , divide coefficients and cancel the common factor $x$ to get $3 y$ . Cancelling means dividing numerator and denominator by the same nonzero factor.

The original denominator cannot be zero. This example requires $x \neq 0$ , even after simplification hides $x$ . Cancel factors, not pieces of a sum: $x + 3$ does not have a factor $x$ in every term.

A common trap

Cancelling $x$ from $\frac{x + 4}{x}$ and writing $4$ .

Instead: The numerator is a sum, not a product with common factor $x$ . At $x = 2$ , the quotient is $3$ , not $4$ .

02 / WORKED EXAMPLES

See the thinking

Example 1

A repeated variable

Simplify $(3 x) (- 4 x)$ .

Coefficients: $3 (- 4) = - 12$ .
Variables: $x \times x = x^{2}$ .

- 12 x^{2}

Example 2

A quotient of terms

Simplify $\frac{18 a^{2} b}{6 a b}$ for $a \neq 0, b \neq 0$ .

Divide coefficients: $18 / 6 = 3$ .
Cancel one factor $a$ , leaving one $a$ .
Cancel the factor $b$ .

3 a

Check your understanding

Simplify $(2 a) (5 a)$ .

Show answer

$10 a^{2}$

$2 \times 5 = 10$ and $a \times a = a^{2}$ .

03 / WORKSHEET

Your turn

6 questions · 13 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Simplify $(4 x) (3 y)$ .
2 marks
Simplify $(- 2 a) (6 a)$ .
2 marks
Simplify $\frac{21 x y}{7 x}$ for $x \neq 0$ .
2 marks
Simplify $\frac{24 m^{2}}{6 m}$ for $m \neq 0$ .
2 marks
Simplify $(3 p) (- 2 q) (2)$ .
2 marks
Why does $\frac{8 t}{2 t} = 4$ require $t \neq 0$ ?
3 marks

Read the worked solutions

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

Simplify $(4 x) (3 y)$ .
2 marks
Multiply coefficients and retain both variable factors.
$12 x y$
Simplify $(- 2 a) (6 a)$ .
2 marks
$- 2 \times 6 = - 12$ and $a \times a = a^{2}$ .
$- 12 a^{2}$
Simplify $\frac{21 x y}{7 x}$ for $x \neq 0$ .
2 marks
Divide $21$ by $7$ and cancel $x$ .
$3 y$
Simplify $\frac{24 m^{2}}{6 m}$ for $m \neq 0$ .
2 marks
Divide coefficients to get $4$ .
Cancel one common factor $m$ .
$4 m$
Simplify $(3 p) (- 2 q) (2)$ .
2 marks
Numerical product: $3 (- 2) (2) = - 12$ .
Keep $p$ and $q$ .
$- 12 p q$
Why does $\frac{8 t}{2 t} = 4$ require $t \neq 0$ ?
3 marks
Cancellation divides by $t$ .
Division by zero is not allowed.
At $t = 0$ , the original denominator is zero and the quotient is undefined.

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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 ↗

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