Expanding brackets

Algebra: Expanding brackets

Expanding brackets

We often see brackets in mathematical expressions. It is, therefore, useful to know how to expand those brackets.

Example

\[\begin{array}{rcl}\blue2 (\green{x}+\purple{2})&=&\blue{2} \green{x} + \blue{2}\cdot \purple{2}
\\ &=& \blue{2} \green{x} + 4\end{array}\]

Minus signs

Pay attention to the minus signs. Remember that for multiplication, the following is true:

\[\begin{array}{r|l}
+\cdot + = + \qqquad & \qqquad - \cdot - = + \\
+ \cdot - = - \qqquad & \qqquad - \cdot + = -
\end{array}\]

Example

\[\begin{array}{rcl}{{-}2} ({{-}5}+{7x})&=&{{-}2} \cdot {{-}5} + {{-}2} \cdot {7x}\\&=&10{-}14x\end{array}\]

Brackets with multiple terms

The rules above are formulated for cases with two terms inside the brackets, but the rule also applies if there are multiple terms inside the brackets.

\[\blue{a} \left(\green{b}+\purple{c}+\color{purple}{d}\right)=\blue{a} \green{b} + \blue{a} \purple{c} + \blue{a} \color{purple}{d}\]

\[\blue{a} \left(\green{b}+\purple{c}+\color{purple}{d}+\orange{e}\right)=\blue{a} \green{b} + \blue{a} \purple{c} + \blue{a} \color{purple}{d}+\blue{a} \orange{e}\]

Example

\[\begin{array}{rcl}(\blue{2}+\green{y}) \purple{x}&=& \blue{2} \purple{x} + \green{y} \purple{x} \\
&=& \blue{2} \purple{x} + \purple{x}\green{y} \end{array}\]

Write without brackets and simplify as much as possible: #-7 \cdot \left(c+9\right)#.

#-7\cdot c -63#

#\begin{array}{rcl}
-7 \cdot \left(c+9\right)&=& -7 \cdot c -7 \cdot 9
\\ &&\phantom{xxx}\blue{\text{rule \(a \cdot \left(b+c\right) = a \cdot b + a \cdot c\) }}\\
&=& -7\cdot c -63
\\ &&\phantom{xxx}\blue{\text{multiplied }}\\
\end{array}#

New example