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