Question & Answer

Factorise the following algebraic expression: ${a^3} - {b^3} - a + b$.

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Hint: First derive the algebraic identity for the difference of cubes of two variables using the cube of difference of two variables. Then try to use this algebraic formula to factorize the expression by taking out a common factor.

Complete step-by-step answer:

In mathematics, an identity is an equality relating one mathematical expression to another mathematical expression, such that both the expressions (which might contain some variables) produce the same value for all values of the variables within a certain range of validity.

Certain identities form the basis of algebra which are known as algebraic identities. They are used to simplify other algebraic expressions.
In the problem, we are given the expression:
${a^3} - {b^3} - a + b{\text{ (1)}}$
We need to use basic addition, subtraction and multiplication identities to simplify this expression into factors.
First let us split the ${a^3} - {b^3}$ part of expression (1) into factors.
We know that,
\[{\left( {a - b} \right)^3} = {a^3} - {b^3} - 3ab\left( {a - b} \right)\]
Transposing the third term on RHS to LHS, we get,
\[ \Rightarrow {\left( {a - b} \right)^3} + 3ab\left( {a - b} \right) = {a^3} - {b^3}\]
Taking $(a - b)$ as a common factor on LHS, we get
\[ \Rightarrow \left( {a - b} \right)\left( {{{\left( {a - b} \right)}^2} + 3ab} \right) = {a^3} - {b^3}\]
Using \[{\left( {a - b} \right)^2} = {a^2} + {b^2} - 2ab\] in above, we get
   \Rightarrow \left( {a - b} \right)\left( {{a^2} + {b^2} - 2ab + 3ab} \right) = {a^3} - {b^3} \\
   \Rightarrow {a^3} - {b^3} = \left( {a - b} \right)\left( {{a^2} + {b^2} + ab} \right){\text{ (2)}} \\
Now since ${a^3} - {b^3}$ is factorised, using equation (2) in equation (1), we get,
$ \Rightarrow {a^3} - {b^3} - a + b = \left( {a - b} \right)\left( {{a^2} + {b^2} + ab} \right) - \left( {a - b} \right)$
Taking $(a - b)$ as a common factor in the above expression, we get
$ \Rightarrow {a^3} - {b^3} - a + b = \left( {a - b} \right)\left( {{a^2} + {b^2} + ab - 1} \right)$
Hence the value of ${a^3} - {b^3} - a + b$ in factored form is $\left( {a + b} \right)\left( {{a^2} + {b^2} + ab - 1} \right)$.

Note: Factorization or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind. Try to remember and use the algebraic identities to simplify and factorize the expressions like above. Effort should be made to form a common factor throughout the expression.