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What is the value of the square of the binomial $a - b$, or what is ${\left( {a - b} \right)^2} = $?

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Last updated date: 23rd Jul 2024
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Answer
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Hint: Split the expression into two binomials and add the like terms.

In the given problem we need to find the value of the square of the binomial $a - b$ .
${\left( {a - b} \right)^2}{\text{ (1)}}$
Splitting the expression $(1)$ into two binomials, we get
${\left( {a - b} \right)^2} = \left( {a - b} \right) \times \left( {a - b} \right)$
Further multiplying the two binomials, we get
$\left( {a - b} \right) \times \left( {a - b} \right) = {a^2} - ba - ab + {b^2}{\text{ (2)}}$
Since multiplication is a commutative operation,
$ \Rightarrow ab = ba$
Using the above relation in equation $(2)$ and adding the like terms, we get
$\left( {a - b} \right) \times \left( {a - b} \right) = {a^2} - 2ab + {b^2}{\text{ (3)}}$
Form equations $(1)$ and $(3)$ we get
${\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}$ , which is the required answer.
Note: It is advised to remember the above found result as it can be used as an identity to solve the larger polynomial expressions.