Factoring a Sum or Difference of Two Cubes
Factor: 27w^{3} + 125y^{12}
Solution
Step 1 Decide if the given polynomial fits a pattern.
The first term, 27w^{3}, is a perfect cube, (3w)^{3}.
The last term, 125y^{12}, is a perfect cube, (5y^{4})^{3}.
The terms are added.
Therefore, 27w^{3} + 125y^{12} is a sum of two cubes.
Step 2 Identify a and b. Then substitute in the pattern and simplify.
In the factoring pattern for a sum of two cubes,
substitute 3w for a and 5y^{4} for b.
Simplify. 
a^{3} + b^{3}
(3w)^{3}  (5y^{4})^{3} 
= (a + b)(a^{2}  ab + b^{2})
= (3w  5y^{4})[(3w)^{2}  3w Â·
5y^{4}
+ (5y^{4})^{2}]
= (3w  5y^{4})(9w^{2}  15y^{4} + 25y^{8}) 
The result is: 27w^{3} + 125y^{12} = (3w  5y^{4})(9w^{2}  15y^{4} + 25y^{8}).
You can multiply to check the factorization. We leave the check to you.
