How do you factor ${a^3} + {b^3}?$

Hint: There is no specific method to factorize the given expression. So, to find the factors or to factorize the expression, remember any formula which has similar terms as present in the given. And then try to fit the given expression in that identity and finally factorize the expression.

Formula used:
Algebraic identity for cube of sum of two numbers: \[{(x + y)^3} = {x^3} + 3{x^2}y + 3x{y^2} + {y^3}\]
Algebraic identity for cube of sum of two numbers: ${(a + b)^2} = {a^2} + 2ab + {b^2}$

Complete step-by-step solution:
In order to factorize the given expression ${a^3} + {b^3}$ we will use the algebraic identity for cube of sum of two numbers, which is given as following
\[\Rightarrow {(x + y)^3} = {x^3} + 3{x^2}y + 3x{y^2} + {y^3}\]
We can simplify and write this further as,
\[\Rightarrow {x^3} + {y^3} = {\left( {x + y} \right)^3} - \left( {3{x^2}y + 3x{y^2}} \right)\]
Now, coming to the question, and fitting our expression in the algebraic identity as $x = a\;{\text{and}}\;y = b$ we will get
\[\Rightarrow {a^3} + {b^3} = {\left( {a + b} \right)^3} - \left( {3{a^2}b + 3a{b^2}} \right)\]
Now we can see that in the right side of the right hand side expression we can take $3ab$ common, so taking it common we will get
\[\Rightarrow {\left( {a + b} \right)^3} - 3ab\left( {a + b} \right)\]
Now again taking $(a + b)$ common from the above expression, we will get
\[\Rightarrow \left( {a + b} \right)\left( {{{\left( {a + b} \right)}^2} - 3ab} \right)\]
Here we get the two factors, but still we should simplify the second one, by using the algebraic identity of square of sum of two numbers, that is ${(a + b)^2} = {a^2} + 2ab + {b^2}$, we will get
\[\Rightarrow \left( {a + b} \right)\left( {{a^2} + 2ab + {b^2} - 3ab} \right)\]
Now performing the respective algebraic operation between similar terms, we will get
\[
   \Rightarrow \left( {a + b} \right)\left( {{a^2} + 2ab - 3ab + {b^2}} \right) \\
   \Rightarrow \left( {a + b} \right)\left( {{a^2} - ab + {b^2}} \right) \\
 \]

So \[\left( {a + b} \right)\;{\text{and}}\;\left( {{a^2} - ab + {b^2}} \right)\] are the required factors of the expression ${a^3} + {b^3}$

Note: It may be difficult to remember every formula of algebra. So, all you can do is just remember a few basic one of them so that you can derive any other required formula at any time in no time. And use them to simplify your calculations and solutions.