Thread: Factoring Sum and Difference of Cubes

There are so many different lectures on how to factor the sum and difference of cubes.

What is the easiest method to use?

For example:

How do you factor the following two questions:

(1) x^6 - 1

(2) x^3 - 125

so many? only two I know of ...

$\displaystyle a^3 + b^3 = (a + b)(a^2 - ab + b^2)$

$\displaystyle a^3 - b^3 = (a - b)(a^2 + ab + b^2)$


$\displaystyle x^6 - 1 = (x^2)^3 - 1^3$ see the "a" and "b" ?

$\displaystyle x^3 - 125 = x^3 - 5^3$ ditto?

now use the factoring pattern for
$\displaystyle a^3 - b^3$.

Hello, magentarita!


There is a trick for memorizing the cube-factoring.


There are two forms:

. . $\displaystyle \begin{array}{cccc}\text{Sum of cubes:} & a^3+ b^3 &=& (a+b)\,(a^2-ab+ b^2) \\
\text{Diff. of cubes:} & a^3-b^3 &=& (a-b)\,(a^2 + ab + b^2) \end{array}$

They can be combined: .$\displaystyle a^3 \pm b^3 \;=\;(a \pm b)\,(a^2 \mp ab + b^2)$



We must memorize the letters: . $\displaystyle a^3 \qquad b^3 \;\;=\;\;(a\qquad b)\,(a^2 \qquad ab \qquad b^2)$


To place the signs, remember the word SOAP:

. . . . . . . . . . . . . . .Opposite
. . . . . . . . . . . . . . . . .$\displaystyle \downarrow$
. . $\displaystyle a^3 \;{\color{red}\pm} \;b^3 \;\;=\;\; (a \;{\color{red}\pm} \;b)\,(a^2 \;{\color{red}\mp} \;ab \;{\color{red}+} \;b^2)$
. . . . . . . . . . . .$\displaystyle \uparrow\qquad \qquad \quad\;\:\uparrow $
. . . . . . . . . . Same . . . . Always Positive