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Extend and apply the distributive law to the expansion of algebraic expressions (ACMNA190)


Source: Australian Curriculum, Assessment and Reporting Authority (ACARA)

Expanding brackets

Numbers obey the distributive laws for multiplication over addition and subtraction. For example:

\begin{align*} &3 \times(4+5)=3\times 4+3\times 5 \hspace{12mm} 7\times (6-3)=7 \times 6-7 \times 3\\\\ &(6 + 8)\times 3=6\times 3+8\times3 \end{align*}

The right distributive laws for division over addition and subtraction also hold as shown. 
For example:

\[ (8+6)\div 2=8\div 2+6\div 2\hspace{12mm} \text{and}\hspace{12mm} \dfrac{9-7}{3}=\dfrac{9}{3}-\dfrac{7}{3} \]

Note that the distributive law for division is valid only in this case, when the divisor is on the right of the bracket.

As with adding like terms and multiplying terms, the laws that apply to arithmetic can be extended to algebra. This process of rewriting an expression to remove brackets is usually referred to as expanding brackets.