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