Multiplying/dividing algebraic terms | Year 8

01 / LEARN

Understand the idea.

Multiply factors

An algebraic term is a product of its coefficient and variable factors. In $(3 a) (4 b)$ , multiply $3 \times 4$ and $a \times b$ to obtain $12 a b$ . A repeated factor gives a power, so $(3 x) (4 x) = 12 x^{2}$ .

Multiplication and addition do different jobs. The sum $3 x + 4 x$ is $7 x$ , while their product is $12 x^{2}$ . Keep a negative coefficient in brackets when multiplying, and determine the sign as well as the numerical coefficient.

Divide by common factors

To divide terms, write a fraction and identify factors shared by its numerator and denominator. In $\frac{15 x y}{5 x}$ , the coefficient quotient is $3$ and the shared x factor cancels, leaving $3 y$ . Cancellation means division by a nonzero factor.

Keep restrictions from the original denominator even if its variables disappear. Here $x \neq 0$ remains required. Only whole factors cancel: the x in $x + 3$ is a term of a sum, so it cannot simply cancel against a denominator x.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

A repeated variable

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

Worked solution

1
Separate coefficients and variable factors.
$(3 x) (- 4 x) = (3 (- 4)) (x \times x)$
Reordering factors preserves a product and makes the two calculations visible.
2
Multiply the coefficients.
$3 (- 4) = - 12$
A positive coefficient times a negative coefficient gives a negative product.
3
Combine the repeated variable factors.
$\begin{aligned} x \times x & = x^{2} \\ (3 x) (- 4 x) & = - 12 x^{2} \end{aligned}$
Two factors of the same variable are represented by a square.

Answer

$- 12 x^{2}$

EXAMPLE 02

A quotient of terms

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

Worked solution

1
Separate the common factors.
$\frac{18 a^{2} b}{6 a b} = \frac{18}{6} \times \frac{a^{2}}{a} \times \frac{b}{b}$
The denominator is a product and is nonzero for the stated a and b values.
2
Simplify each quotient.
$\begin{aligned} \frac{18}{6} & = 3 \\ \frac{a^{2}}{a} & = a \\ \frac{b}{b} & = 1 \end{aligned}$
One a factor and the b factor cancel by division by nonzero quantities.
3
Multiply the remaining factors.
$\frac{18 a^{2} b}{6 a b} = 3 a, a \neq 0, b \neq 0$
The simplified expression keeps the restrictions of the original quotient.

Answer

$3 a$

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

$10 a^{2}$

Multiply coefficients and variable factors separately: $2 \times 5 = 10$ and $a \times a = a^{2}$ . Hence $(2 a) (5 a) = 10 a^{2}$ .

Keep practising

6 worksheet questions · 13 marks

Open worksheet 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 ↗

In this lesson

Understand the idea.

Multiply factors

Divide by common factors

See the method in action.

A repeated variable

Separate coefficients and variable factors.

Multiply the coefficients.

Combine the repeated variable factors.

A quotient of terms

Separate the common factors.

Simplify each quotient.

Multiply the remaining factors.

Try it yourself.

Check your understanding

Keep practising