Answer
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Hint: To solve such fraction we have to either factorize the given expression with standard set of rules or if factorization is not possible then we can expand the term, by using standard algebraic properties, here for this question we have to expand the given term and then the output we get will be our final solution.
Formulae Used:
\[{a^3} - {b^3} = (a - b)({a^2} + ab + {b^2})\]
Complete step by step solution:
The given question is \[{x^3} - 1\]. To solve this question we have to use the standard property of algebra which allows the expansion of the cubic expression for two terms, the demanded property which have to be used here is :
\[ \Rightarrow {a^3} - {b^3}\]
The expansion of the above term can be described as:
\[ \Rightarrow {a^3} - {b^3} = (a - b)({a^2} + ab + {b^2})\]
The above expression can be cross verified by solving the right hand side of the equation.
Now, applying this to our given question, first we have to make our term looks like the given expression, on solving we get:
\[\Rightarrow {x^3} - 1 \\
\Rightarrow {x^3} - {1^3}\]
Here we have done the arrangement like we have put the power of three on the constant term one, and providing any positive or negative power to one except zero does not change the value of one, now applying the formulae we get:
\[ \therefore{x^3} - {1^3} = (x - 1)({x^2} + x \times 1 + {1^2}) = (x + 1)({x^2} + x + 1)\]
Note: For this type of expression where factorization cannot be found by mid-term splitting or direct taking common, then we have to modify the question and make it look like the standard expression to simplify and get the solution for the expression given.
Formulae Used:
\[{a^3} - {b^3} = (a - b)({a^2} + ab + {b^2})\]
Complete step by step solution:
The given question is \[{x^3} - 1\]. To solve this question we have to use the standard property of algebra which allows the expansion of the cubic expression for two terms, the demanded property which have to be used here is :
\[ \Rightarrow {a^3} - {b^3}\]
The expansion of the above term can be described as:
\[ \Rightarrow {a^3} - {b^3} = (a - b)({a^2} + ab + {b^2})\]
The above expression can be cross verified by solving the right hand side of the equation.
Now, applying this to our given question, first we have to make our term looks like the given expression, on solving we get:
\[\Rightarrow {x^3} - 1 \\
\Rightarrow {x^3} - {1^3}\]
Here we have done the arrangement like we have put the power of three on the constant term one, and providing any positive or negative power to one except zero does not change the value of one, now applying the formulae we get:
\[ \therefore{x^3} - {1^3} = (x - 1)({x^2} + x \times 1 + {1^2}) = (x + 1)({x^2} + x + 1)\]
Note: For this type of expression where factorization cannot be found by mid-term splitting or direct taking common, then we have to modify the question and make it look like the standard expression to simplify and get the solution for the expression given.