The First and Second Algebraic Identity

You may have previously seen that ${(a\times b)}^2=a^2\times b^2$, but what is ${(a+b)}^2$? If you write ${(a+b)}^2$ as $(a+b)(a+b)$, you can use what you learned about the distributive property of multiplying parentheses to find the answer:

$$\begin{array}{llll}\hfill {(a+b)}^2& =(a+b)(a+b)& \hfill & \\ \hfill & =a^2+ab+ba+b^2& \hfill & \\ \hfill & =a^2+2ab+b^2& \hfill & \end{array}$$

This also applies to ${(a-b)}^2$:

$$\begin{array}{llll}\hfill {(a-b)}^2& =(a-b)(a-b)& \hfill & \\ \hfill & =a^2-ab-ba+b^2& \hfill & \\ \hfill & =a^2-2ab+b^2& \hfill & \end{array}$$

These results are used all the time in mathematics, and we call them the first and second algebraic identities. Learn these, and you’ll save a lot of time you would’ve spent calculating!