Question

How do you multiply $(a - b)(a - b)$ ?

Answer 1

We need to simplify the given algebraic expression. The algebraic expression is given in the form of the product of two binomials. So we will simplify it by multiplying both the factors with each other and write it in simplified form.

Formula used: We will use the identity, ${(a - b)^2} = {a^2} - 2ab + {b^2}$

Complete step-by-step answer:
We need to multiply the given expression, which is written in the form of the product of two binomials. In other words, we say that the expression is written in the form of the product of two factors.
We know that a binomial is a polynomial containing two terms.
The given algebraic expression is $(a - b)(a - b)$,
$= a(a - b) - b(a - b)$, we multiply the first term of the first factor with the second factor and the second term of the first factor with the second factor then this becomes,
$= {a^2} - ab - ab + {b^2}$
$= {a^2} - 2ab + {b^2}$, which is a trinomial.

This is also called a perfect square trinomial, as it is obtained on squaring a binomial, i.e., ${(a - b)^2} = {a^2} - 2ab + {b^2}$.

Additional information: An identity is an equation that is true for all values of the variables. Here, the identity is ${(a - b)^2} = {a^2} - 2ab + {b^2}$. We can verify the identity by putting $0$, $- 1$, $1$, $2$, etc. at the place of variables. Then, we will find that the left side remains equal to the right side for all values of the variables.

Note:
There is an alternate way to multiply the given expression. The given expression is $(a - b)(a - b)$, this can also be written as the square of the factor like, ${(a - b)^2}$ . Then we can directly apply the identity ${(a - b)^2} = {a^2} - 2ab + {b^2}$ .