How do you divide \[\dfrac{{{x^{^3}} - 1}}{{x - 1}}\] ?

Hint: In the above question, is based on simplifying the algebraic expression. The main approach towards this question is to apply the formula of difference of cubes in the numerator. For that every term in the numerator should be a perfect cube and then further the common terms get cancelled in the numerator and denominator.

Complete step-by-step answer:
Algebraic expression in mathematics is an expression which is made up of variables, constants along with mathematical operations like addition, subtraction, division and multiplication. So, the combination of terms is said to be an expression.
We need to simplify the above expression. It can be simplified by first applying the formula of difference of cubes.
The formula for difference of cubes is as follows:
 \[{a^3} - {b^3} = (a - b)({a^2} + ab + {b^2})\]
We can apply this formula on the numerator by writing the terms in the numerator in the form of cubes.
So, we can write the first term as \[{\left( x \right)^3}\] and we can write the second term as \[{\left( 1 \right)^3}\] .It can be written as
 \[{\left( x \right)^3} - {\left( 1 \right)^3}\]
Further by solving we get,
 \[
  \dfrac{{{{\left( x \right)}^3} - {{\left( 1 \right)}^3}}}{{x - 1}} \\
   = \dfrac{{\left( {x - 1} \right)\left( {{x^2} + x + 1} \right)}}{{\left( {x - 1} \right)}} \;
 \]
We can see that there are two terms in the numerator and denominator which is common. Therefore, by cancelling it we get
 \[{x^2} + x + 1\]
So, we get the above expression after simplifying it.
So, the correct answer is “ \[{x^2} + x + 1\] ”.

Note: An important thing to note is that whenever we apply the sum or difference of cubes, we need to make sure the terms in the expression are perfect cubes of a number. For example: 8 is a perfect cube of the number 2.