Expanding double brackets

Algebra: Expanding brackets

Expanding double brackets

We have seen how we can expand single brackets. We can also expand double brackets. We can do so using the banana method, as they call it in Dutch.

The banana method

Example

Relation to single brackets

As the banana-shaped arches already indicate, the right-side of the banana formula is created by expanding brackets twice: \[\begin{array}{rcl}(\blue{a}+\green{b})(\purple{c}+\orange{d}) &=& \blue{a} (\purple{c}+\orange{d}) + \green{b} (\purple{c}+\orange{d})\\ &=& \blue{a} \purple{c}+\blue{a} \orange{d} +\green{b} \purple{c} +\green{b} \orange{d}\end{array}\] Sometimes it can be useful to use a multiplication chart: \[\begin{array}{c|c|c} & \purple{c} & \orange{d}\\ \hline \blue{a} & \blue{a} \purple{c} & \blue{a} \orange{d}\\ \hline \green{b} & \green{b} \purple{c} & \green{b} \orange{d} \end{array}\] After completing the chart, you add the four products together.

Mnemonic

A way to remember the method is the mnemonic FOIL, which stands for First, Outer, Inner, Last. This corresponds to the order of the operations.

First, multiply the first terms in both brackets,
then the first term in the first brackets with the last term in the second brackets, i.e. the 'outer' terms,
then the last of the first brackets with the first of the last brackets, i.e. the 'inner' terms,
and finally, the last terms in both brackets.

Example

#\left( \blue a + \green b \right) \cdot \left( \purple c + \orange d \right) = #
##
#\blue a \cdot \purple c \; +#
#\blue a \cdot \orange d \; +#
##
#\green b \cdot \purple c \; +#
##
#\green b \cdot \orange d \; \phantom{+}#

In the same way, we can also work with more complicated products.

Expand the brackets and simplify as much as possible: #\left(b-9\right) \cdot \left(b + 9 \right)# .

#b^2-81#

#\begin{array}{rcl}
\left(b-9\right) \cdot \left(b + 9 \right) &=& b \cdot b + b \cdot 9 -9 \cdot b -9 \cdot 9 \\ &&\phantom{xxx}\blue{\text{rule \(\left(a+b\right) \cdot \left(c+d\right) = a \cdot c + a \cdot d + b \cdot c + b \cdot d\) }}\\
&=& b^2 + 9\cdot b -9\cdot b -81
\\&&\phantom{xxx}\blue{\text{multiplied }}\\
&=& b^2-81
\\&&\phantom{xxx}\blue{\text{simplified}}\\
\end{array}#

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