Question

The value of $\left( a+b \right)\left( a-b \right)$ is
A) ${{a}^{2}}-{{b}^{2}}$
B) ${{\left( a+b \right)}^{2}}$
C) ${{\left( a-b \right)}^{2}}$
D) None of these

Answer 1

Hint:
To find the value of the above expression, we will use the distributive property of multiplication. We will first consider the whole terms inside the bracket as a single term and then we will apply the distributive property of multiplication and then again we will apply the distributive property for the left terms to simplify it further.

Complete step by step solution:
We need to find the value of algebraic expression $\left( a+b \right)\left( a-b \right)$
We will first consider $\left( a+b \right)$ as a single term and then we will apply the distributive property of multiplication.
Applying distributive property of multiplication here, we get
$=\left( a+b \right)\times a-\left( a+b \right)\times b$
Now, we will again apply the distributive property of multiplication for the terms $\left( a+b \right)\times a$ and the terms $\left( {a + b} \right) \times b$.
$=a\times a+b\times a-\left[ a\times b+b\times b \right]$
Now, we will multiply -1 to all the terms present inside the bracket.
$=a\times a+b\times a-a\times b-b\times b$
Multiplying the terms, we get
$={{a}^{2}}+b\times a-a\times b-{{b}^{2}}$
Subtracting all the like terms, we get
$={{a}^{2}}-{{b}^{2}}$

Thus, the value of the given algebraic expression is equal to ${{a}^{2}}-{{b}^{2}}$

Note:
Since we have used the distributive property of multiplication to multiply the terms. Distributive property of multiplication states that if $a$, $b$ and $c$ are three numbers, then $a\left( b+c \right)=a\times b+a\times c$ .
After solving the given problem, we have got $\left( a+b \right)\left( a-b \right)={{a}^{2}}-{{b}^{2}}$, this is one of the important algebraic identities that is used directly in the problems which are complicated and lengthy, so as to make them easy to solve. Here algebraic identities are defined as equalities which are true and valid for all values of the variable. Also, the algebraic identities are used to factorize the polynomials. Always keep in mind that all the algebraic identities are derived from the Binomial Theorem.