How do you simplify and divide \[\left( {{x^3} + {y^3}} \right) \div \left( {x + y} \right)\] ?

Hint: This is a very simple question to solve. Here we will just expand the identity \[{x^3} + {y^3}\] and then divide as per question. Simplifying the expression is nothing but cancelling the terms like terms if observed and performing necessary operations. We know that \[\left( {{x^3} + {y^3}} \right) = \left( {x + y} \right)\left( {{x^2} - xy + {y^2}} \right)\] . Using this expansion of identity we will solve the given expression.

Complete step-by-step answer:
Given that
 \[\left( {{x^3} + {y^3}} \right) \div \left( {x + y} \right)\]
Now write the expanded form,
 \[\left( {{x^3} + {y^3}} \right) \div \left( {x + y} \right) = \dfrac{{\left( {x + y} \right)\left( {{x^2} - xy + {y^2}} \right)}}{{\left( {x + y} \right)}}\]
Now we observe that the first bracket of numerator is exactly the same as the denominator. So we can cancel them easily.
 \[\left( {{x^3} + {y^3}} \right) \div \left( {x + y} \right) = {x^2} - xy + {y^2}\]
Now if we observe that the remaining equation is neither a quadratic equation nor any standard identity. So the answer completes here.
So, the correct answer is “${x^2} - xy + {y^2}$”.

Note: Note that algebraic identities are the main part of algebraic maths. Here no other mathematical tools are required. Simply writing the expansion of identity will help to write the answer. There are other algebraic identities also. That is having square and cubic expansion. Unlike algebraic there are trigonometric identities also there.
Remember that \[{x^3} + {y^3}\] and \[{\left( {x + y} \right)^3}\] are two different identities. Sometimes we get confused and the expansion we write is also messed up.