How do you factor \[{x^5} - {y^5}\]?

Hint: Here, we will find the factors of the algebraic expression by using the algebraic identity of the difference between the numbers of the \[{n^{th}}\]power. We will then simplify it further to get the required answer. Factorization is a process of rewriting the expression in terms of the product of the factors.

Complete Step by Step Solution:
We are given that \[{x^5} - {y^5}\].
We will find the factors for the given algebraic expression by using the algebraic identity.
The difference between the numbers of the \[{n^{th}}\] power is given by the Algebraic Identity: \[{x^n} - {y^n} = \left( {x - y} \right)\left( {{x^{n - 1}} + {x^{n - 2}}y + ..... + x{y^{n - 2}} + {y^{n - 1}}} \right)\]
Now, we will find the factors by using the algebraic identity, we get
By substituting \[n = 5\] in the formula, we get
\[{x^5} - {y^5} = \left( {x - y} \right)\left( {{x^{5 - 1}} + {x^{5 - 2}}{y^{5 - 4}} + {x^{5 - 3}}{y^{5 - 3}} + {x^{5 - 4}}{y^{5 - 2}} + {y^{5 - 1}}..... + x{y^{n - 2}} + {y^{n - 1}}} \right)\]
By simplifying the equation, we get
\[ \Rightarrow {x^5} - {y^5} = \left( {x - y} \right)\left( {{x^4} + {x^3}y + {x^2}{y^2} + x{y^3} + {y^4}} \right)\]
In the above expression, there are no common terms, so we cannot factorize it further.

Therefore, the factors of \[{x^5} - {y^5}\] is \[\left( {x - y} \right)\left( {{x^4} + {x^3}y + {x^2}{y^2} + x{y^3} + {y^4}} \right)\].

Note: We know that factors are numbers if the expression is a numeral. Factors are algebraic expressions if the expression is an algebraic expression. Factorization is done by using the common factors, the grouping of terms, and the algebraic identity. We know that an equality relation that is true for all the values of the variables is called an Identity. We should be careful that the algebraic expression has to be rewritten in the form of algebraic identity, if any factor is common, then it can be taken out as a common factor and check whether all the terms after taking common factor is in the form of Algebraic Identity.