Factorize the given algebraic expression: $a-b-{{a}^{2}}+{{b}^{2}}$
(a) $\left( a-b \right)\left[ 1-\left( a+b \right) \right]$
(b) $\left( a+b \right)\left[ 1-\left( a-b \right) \right]$
(c) $\left( a-b \right)\left[ 1-\left( a-b \right) \right]$
(d) None of these

Hint: Simplify the given algebraic expression by using the algebraic identity ${{x}^{2}}-{{y}^{2}}=\left( x+y \right)\left( x-y \right)$. Further, simplify the expression by taking out the common terms to completely factorize the given expression into linear factors.

Complete step-by-step solution -
We have to factorize the algebraic expression $a-b-{{a}^{2}}+{{b}^{2}}$. To do so, we will first simplify the given expression.
Taking out the common terms, we can rewrite the given algebraic expression as $a-b-\left( {{a}^{2}}-{{b}^{2}} \right)$.
We know the algebraic identity ${{x}^{2}}-{{y}^{2}}=\left( x+y \right)\left( x-y \right)$.
We can simplify the expression $a-b-\left( {{a}^{2}}-{{b}^{2}} \right)$ using the above identity as $a-b - \left[ \left( a+b \right)\left( a-b \right) \right]$.
Taking out the common terms from the above expression, we have $\left( a-b \right)\left[ 1-\left( a+b \right) \right]$.
Hence, we have factorized the given algebraic expression $a-b-{{a}^{2}}+{{b}^{2}}$ into linear factors as $\left( a-b \right)\left[ 1-\left( a+b \right) \right]$, which is option (a).

Note: We must know the difference between an algebraic identity and an algebraic expression. An algebraic identity is an equality that holds for all possible values of its variables. We can prove each of the identities by performing basic algebraic operations such as addition, multiplication, subtraction, and division. They are used for the factorization of the polynomials. That’s why they are useful in the computation of algebraic expressions. An algebraic expression differs from an algebraic identity in the way that the value of algebraic expression changes with the change in variables. However, an algebraic identity is equality which holds for all possible values of variables.