#### notes

Now study the three identiies which are very useful in our work.

These identities are obtained by multiplying a binomial by another binomial.

Let us first consider the product `(a + b)^2 = (a + b) (a + b)`

= a(a + b) + b (a + b)

= `a^2 + ab + ba + b^2`

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

Thus , **`(a + b)^2 = a^2 + 2ab + b^2`** (I)

⦁ we consider `(a – b)^2 = (a – b) (a – b)

= a (a – b) – b (a – b)

= `a^2 – ab – ba + b^2 = a^2 – 2ab + b^2`

or **`(a – b)^2 = a^2 – 2ab + b^2`** (II)

⦁ Finally, consider (a + b) (a – b).

We have (a + b) (a – b) = a (a – b) + b (a – b)

= `a^2 – ab + ba – b^2 = a^2 – b^2 ` (since ab = ba)

or **`(a + b) (a – b) = a^2 – b^2`** (III)

The identities (I), (II) and (III) are known as standard identities.