Explore: The Sum and Difference of Cubes Solutions

Factoring patterns occur for the sum and difference of cubes. 
Perfect cubes occur when the base is multiplied by itself three times. 
$4^{3} = 64 {(2 x)}^{3} = 2^{3} x^{3} = 8 x^{3}$
Continue to look for the GCF as the first step of factoring the sum and difference of cubes.

Sum of Cubes

$a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$

Difference of Cubes

$a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$

Example 9

Factor completely.
$8 x^{3} + 125 y^{3}$

Implement

${(2 x)}^{3} = 8 x^{3}, \begin{matrix} {(5 y)}^{3} = 125 y^{3} \end{matrix} (2 x + 5 y) ({(2 x)}^{2} - (2 x \cdot 5 y) + {(5 y)}^{2}) (2 x + 5 y) (4 x^{2} - 10 x y + 25 y^{2})$

Explain

Find the cubed root of the terms.
Substitute values into the sum of cubes rule.
Simplify.

Example 10

Factor completely.
$10 m^{3} - 640$

Implement

$10 (m^{3} - 64) 4^{3} = 64 10 (m - 4) (m^{2} + 4 m + 16)$

Explain

Factor out the GCF.
Find the cubed root of 64.
Find the difference of cubes.

Note

It is helpful to memorize or have a list of common cubes.

$\begin{matrix} 2^{3} = 8, & 3^{3} = 27, & 4^{3} = 64 \\ 5^{3} = 125, & 6^{3} = 216, & 7^{3} = 343 \end{matrix}$