# Factorize expression with sum of cubes

In this chapter we will try to factorize the algebraic expression given in the form of \mathtt{a^{3} +b^{3}}

At the end of the chapter, some solved problems are also given for your practice.

## How to factorize \mathtt{a^{3} +b^{3}} ?

To factorize the expression \mathtt{a^{3} +b^{3}} , use the following formula;

\mathtt{a^{3} +b^{3} =\ ( a+b)\left( a^{2} -ab+b^{2}\right)}

This formula is very important.

Please memorize it as it the questions related to it are directly asked in math exams.

### Problems on \mathtt{a^{3} +b^{3}}

Example 01
Factorize the below expression;

\mathtt{\Longrightarrow \ x^{3} +\ 1331}

Solution
The above expression can be expressed as;

\mathtt{\Longrightarrow \ x^{3} +( 11)^{3}}

Referring the formula;

\mathtt{a^{3} +b^{3} =\ ( a+b)\left( a^{2} -ab+b^{2}\right)}

Where;
a = x
b = 11

Putting the values in the formula, we get;

\mathtt{\Longrightarrow \ ( x\ +11)\left( x^{2} -x.11\ +( 11)^{2}\right)}\\\ \\ \mathtt{\Longrightarrow \ ( x+11)\left( x^{2} -11x\ +121\right)}

Hence, the above expression is the solution.

Example 02
Factorize the below expression

\mathtt{\Longrightarrow 343\ x^{3} +\ 216y^{3}}

Solution
The above expression can be written as;

\mathtt{\Longrightarrow \ ( 7x)^{3} +( 6y)^{3}}

Referring the formula;
\mathtt{a^{3} +b^{3} =\ ( a+b)\left( a^{2} -ab+b^{2}\right)}

Where;
a = 7x
b = 6y

Putting the values in formula;

\mathtt{\Longrightarrow \ ( 7x\ +6y)\left(( 7x)^{2} -( 7x)( 6y) \ +( 6y)^{2}\right)}\\\ \\ \mathtt{\Longrightarrow \ ( 7x+6y)\left( 49x^{2} -42xy\ +36y^{2}\right)}

Hence, the above expression is the solution.

Example 03
Factorize the below expression.

\mathtt{\Longrightarrow 3x^{3} +\ 81}

Solution
The expression can be written as;

\mathtt{\Longrightarrow \ 3\ \left( x^{3} +27\right)}\\\ \\ \mathtt{\Longrightarrow \ 3\ \left( x^{3} +\ 3^{3}\right)}

Referring the formula;
\mathtt{a^{3} +b^{3} =\ ( a+b)\left( a^{2} -ab+b^{2}\right)}

Where;
a = x
b = 3

Using the formula;

\mathtt{\Longrightarrow \ 3\ ( x+3) \ \left( x^{2} +x.3\ +3^{2}\right)}\\\ \\ \mathtt{\Longrightarrow \ 3\ ( x+3)\left( x^{2} +3x+9\right)}

Hence, the above expression is the solution.

Example 04
Factorize the below expression.

\mathtt{\Longrightarrow \ x^{6} +y^{6}}

Solution
The above expression can be written as;

\mathtt{\Longrightarrow \ \left( x^{2}\right)^{3} +\left( y^{2}\right)^{3}}

Using the sum of cubes formula;

\mathtt{\Longrightarrow \left( x^{2} +y^{2}\right)\left(\left( x^{2}\right)^{2} -\left( x^{2}\right)\left( y^{2}\right) \ +\left( y^{2}\right)^{2}\right)}\\\ \\ \mathtt{\Longrightarrow \ \left( x^{2} +y^{2}\right)\left( x^{4} -x^{2} .y^{2} \ +y^{4}\right)}

Hence, the above expression is the solution.

Example 05
Solve the below expression

\mathtt{\Longrightarrow \ 15x^{9} +120y{^{21}}}

Solution
The above expression can be expressed as;

\mathtt{\Longrightarrow \ 15\left(\left( x^{3}\right)^{3} +8\left( y^{7}\right)^{3}\right)}\\\ \\ \mathtt{\Longrightarrow \ 15\left(\left( x^{3}\right)^{3} +\left( 2y^{7}\right)^{3}\right)}

Referring the formula;
\mathtt{a^{3} +b^{3} =\ ( a+b)\left( a^{2} -ab+b^{2}\right)}

Where;
a = \mathtt{x^{3}}
b = \mathtt{2y^{7}}

Using the formula, we get;

\mathtt{\Longrightarrow \ 15\ \left( x^{3} +2y^{7}\right)\left(\left( x^{3}\right)^{2} +x^{3} .2y^{7} +\left( 2y^{7}\right)^{2}\right)}\\\ \\ \mathtt{\Longrightarrow \ 15\left( x^{2} +2y^{7}\right)\left( x^{6} +2x^{3} y^{7} +4y^{14}\right)}

Hence, the above expression is the solution.

Example 06
Find the value of below expression using sum of cube formula.

\mathtt{\Longrightarrow \ 12^{3} +13{^{3}}}

Solution
Applying sum of cube formula;

\mathtt{\Longrightarrow \ ( 12\ +\ 13)\left( 12^{2} -12.13+13^{2}\right)}\\\ \\ \mathtt{\Longrightarrow \ ( 25) .\ ( 144\ -\ 156\ +\ 169)}\\\ \\ \mathtt{\Longrightarrow \ ( 25) .\ ( 157)}\\\ \\ \mathtt{\Longrightarrow \ 3925}

Hence, 3925 is the solution

Next chapter : Problems on factorization of algebraic expression