Explore: The Sum and Difference of Cubes Solutions

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.

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