Question

What patterns are there in the product of the square of a binomial and the product of a sum and a difference?

Answer 1

`\begin{gathered} (a+b)^2=a^2+2ab+b^2 \\ (a+b)(a-b)=a^2-b^2 \end{gathered}`

1) Let's expand the square of a binomial, using "a" for the first term and "b" for the second one:

`(a+b)^2=(a+b)(a+b)=a^2+2ab+b^2`

The square of the first plus twice the product of them plus the square of b.

2) And the product of a sum and a difference:

`(a+b)(a-b)=a^2-ab+ab-b^2=a^2-b^2`

Note that the ab term was canceled out by -ab

The square of the first minus the square of the 2nd term

3) Hence, the patterns are:

`\begin{gathered} (a+b)^2=a^2+2ab+b^2 \\ (a+b)(a-b)=a^2-b^2 \end{gathered}`