The banana method for expanding double brackets

Calculating with letters: Computing with letters

The banana method for expanding double brackets

The banana method

For the product of two two-terms, we have the banana method:

Examples

$\begin{aligned} (a+2)(b+3) &=a\cdot b+a\cdot 3+2\cdot b + 2\cdot 3\\ &= ab+3a+2b+6\\[0.3cm] (-4a+2)(2b-1) &= -4a\cdot 2b+-4a\cdot -1\\ &\phantom{=\,\,} +2\cdot 2b+2\cdot -1\\ &=-8ab+4a+4b-2\end{aligned}$

As the banana-shaped lines indicate, the right-hand side is obtained by applying a distributive property twice: $\begin{aligned}(a+b)(c+d)&= a(c+d) + b(c+d)\\ &= ac +ad + bc + bd\end{aligned}$ Other names for this property are the parrot beak method and the crabs claw method. because of the resemblance with this objects.

Sometimes it can be handy to use a multiplication table: $\begin{array}{c|c|c} \cdot & c & d\\ \hline a & ac & ad\\ \hline b & bc & bd \end{array}$ After completing the table you add the four products.

The multiplication table $\begin{array}{c|c|c|c} \cdot & d & e & f\\ \hline a & ad & ae & af\\ \hline b & bd & be & bf \\ \hline c & cd & ce & cf \end{array}$ illustrates that the product of two expressions with three terms $(a+b+c)(d+e+f)$ can be worked out by multiplying each term within the left pair of brackets and each term within the right pair of brackets, followed by adding all intermediate outcomes: $\begin{aligned}(a+b+c)(d+e+f) \;=\; &\phantom{+}\; ad + ae + af \\ &+\; bd\, + be + bf \\ &+\; cd\, + ce + cf\end{aligned}$

A third method for illustrating the banana method is the area model with rectangles. For example, the formula $(n+2)(n+1)=n^2+3n+1$ can be visualised with rectangles in the following way:

With the banana method you can expand the double pairs of brackets. A square of a two-term can be seen here as the product of two identical two-terms. The formula can also be used in all kinds of complicated situations. Sometimes you can collect terms after expansion of brackets with the banana formula.

Expand all brackets in $(10b-9)(-6b-3)$ .

$\begin{aligned}(10b-9)(-6b-3) &= (10b)\cdot (-6b)+(10b)\cdot (-3)+(-9)\cdot (-6b)+(-9)\cdot (-3)\\ &\phantom{abcdevwxyz}\blue{\text{the banana method}}\\ &= -60b^2-30b+54b+27\\ &\phantom{abcdevwxyz}\blue{\text{simplification of terms}}\\ &=-60b^2+ 24b+27\\ &\phantom{abcdevwxyz}\blue{\text{collection of similar terms}}\end{aligned}$

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