Expansion of of single brackets

Calculating with letters: Computing with letters

Expansion of of single brackets

Distributive properties

The distributive properties of multiplication relative to addition and subtraction (also called distributive laws) are as follows:

$a(b+c)=ab+ac$ and

$(a+b)c=ac+bc$

Examples

$\begin{aligned}3(2a+5)&=3\cdot 2a+3\cdot 5\\ &=6a+15\\ \\ -3a^2(4a-7) &=(-3a^2)\cdot (4a)+(-3a^2)\cdot (-7)\\&=-3\cdot 4a^2\cdot a+-3\cdot -7\cdot a^2\\ &=-12a^3+21a^2\\ \\ (a-3)a&=a\cdot a+(-3)\cdot a\\ &=a^2-3a\end{aligned}$

Minus signs

The properties

$a(b+c)=ab+ac\qquad\mathrm{and}\qquad (a+b)c=ac+bc$ are correct when you replace $a$ and $b$ by numbers and insert the omitted multiplication symbols in the right places, but they remain valid when letters represent algebraic expressions.

Of course you can also formulate distributive properties such as

$a(b-c)=ab-ac\qquad\mathrm{and}\qquad (a-b)c=ac-bc$ but these properties follow immediately from the former properties by reading $-c$ and $-b$ as ${}+(-c)$ and ${}+(-b)$ , respectively.

Multiplication table You can also construct a multiplication table and, after filling out the table , add the two products.

$a(b+c)=ab+ac$	$(a+b)c=ac+bc$
$\begin{array}{c\|c\|c}\cdot & b & c\\ \hline a &ab & ac \end{array}$	$\begin{array}{c\|c} \cdot & c\\ \hline a & ac \\ \hline b & bc \end{array}$

More terms between brackets The distributive properties of multiplication relative to addition and subtraction are defined for the product of a single term with an expression consisting of two terms. But you can also do this for a second expression with three, four, or more terms.

$a(b+c+d)=ab+ac+ad$	$a(b+c+d+e)=ab+ac+ad+ae$
$\begin{array}{c\|c\|c\|c}\cdot & b & c & d\\ \hline a &ab & ac & ad \end{array}$	$\begin{array}{c\|c\|c\|c}\cdot & b & c & d & e\\ \hline a &ab & ac & ad & ae \end{array}$

The distributive properties help you expand brackets in the correct way. Examples illustrate how this goes.

Expand all brackets in $-7(6z+9)$ .

Apply the distributive properties to:

$\begin{aligned}-7(6z+9) &=(-7\times 6z)+(-7\times9)\\&=-42z-63\end{aligned}$

New example

As the examples illustrate, you should be especially cautious when you are dealing with minus signs. An example where things can go wrong:

$-(a^2-a)$ Eliminating the brackets is a little tricky because the one-term is difficult to recognize; you should namely read the expression as

$-1\cdot(a^2-a)$ and then expansion of the brackets goes as follows:

$\begin{aligned}-(a^2-a) &=-1\cdot(a^2-a)\\ &=-1\cdot a^2\;+\;-1\cdot -a\\ &=-a^2+a\end{aligned}$

Mathcentre video

Expanding/Removing Brackets (39:25)