According to the (a+b)(a-b) formula, the $a$ plus $b$ times $a$ minus $b$ is equal to $a$ squared minus $b$ squared.

$(a+b)(a-b)$ $\,=\,$ $a^2-b^2$

In this algebraic identity, $a$ and $b$ are two literals and they are used as variables to represent the terms. The two variables $a$ and $b$ form binomials $a+b$ and $a-b$ by the summation and subtraction respectively. The multiplication of both algebraic expressions is written as follows.

$(a+b) \times (a-b)$

$\implies$ $(a+b)(a-b)$

Now, let’s learn how to prove the $(a+b)(a-b)$ rule in algebraic mathematics by the multiplication of algebraic expressions.

The binomials $a+b$ and $a-b$ are algebraic expressions. So, they can be multiplied by the multiplication of algebraic expressions.

$\implies$ $(a+b)(a-b)$ $\,=\,$ $a \times (a-b)$ $+$ $b \times (a-b)$

The product of two algebraic expressions is transformed into an algebraic expression, which contains two algebraic terms. In each term, the difference basis binomial is multiplied by a variable. The multiplication can be done by the distributive property of multiplication over subtraction.

$\implies$ $(a+b)(a-b)$ $\,=\,$ $a \times a$ $-$ $a\times b$ $+$ $b \times a$ $+$ $b \times (-b)$

The multiplication of the binomial $a-b$ by another binomial $a+b$ is completed and it is time to simplify the right hand side expression of the equation for obtaining the product of the algebraic expressions $a+b$ and $a-b$.

$\implies$ $(a+b)(a-b)$ $\,=\,$ $a^2$ $-$ $ab$ $+$ $ba$ $-$ $b^2$

According to the commutative property of multiplication, the product of $a$ and $b$ is equal to the product of the variables $b$ and $a$. Hence, the expression $ab$ can be written as $ba$ and vice-versa.

$\implies$ $(a+b)(a-b)$ $\,=\,$ $a^2$ $-$ $ab$ $+$ $ab$ $-$ $b^2$

$\implies$ $(a+b)(a-b)$ $\,=\,$ $a^2$ $-$ $\cancel{ab}$ $+$ $\cancel{ab}$ $-$ $b^2$

$\,\,\,\therefore\,\,\,\,\,\,$ $(a+b)(a-b)$ $\,=\,$ $a^2-b^2$

Therefore, it is proved algebraically that $a$ plus $b$ times $a$ minus $b$ is equal to $a$ squared minus $b$ squared. Thus, the $a$ plus $b$ times $a$ minus $b$ algebraic identity is proved in algebraic method by multiplying the algebraic expressions.

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