Sum of two cubes

The mnemonic "SOAP", standing for "Same, Opposite, Always Positive", is sometimes used to memorize the correct placement of the addition and subtraction symbols while factorizing cubes.[2] When applying this method to the factorization, "Same" represents the first term with the same sign as the original expression, "Opposite" represents the second term with the opposite sign as the original expression, and "Always Positive" represents the third term and is always positive.

	original sign		Same		Opposite		Always Positive
${\displaystyle a^{3}}$	${\displaystyle +}$	${\displaystyle b^{3}\quad =\quad (a}$	${\displaystyle +}$	${\displaystyle b)(a^{2}}$	${\displaystyle -}$	${\displaystyle ab}$	${\displaystyle +}$	${\displaystyle b^{2})}$
${\displaystyle a^{3}}$	${\displaystyle -}$	${\displaystyle b^{3}\quad =\quad (a}$	${\displaystyle -}$	${\displaystyle b)(a^{2}}$	${\displaystyle +}$	${\displaystyle ab}$	${\displaystyle +}$	${\displaystyle b^{2})}$

Starting with the expression, ${\displaystyle a^{2}-ab+b^{2}}$ is multiplied by a and b[1]

$${\displaystyle (a+b)(a^{2}-ab+b^{2})=a(a^{2}-ab+b^{2})+b(a^{2}-ab+b^{2}).}$$

By distributing a and b to ${\displaystyle a^{2}-ab+b^{2}}$, one get[1]

$${\displaystyle a^{3}-a^{2}b+ab^{2}+ba^{2}-ab^{2}+b^{3}}$$

and by canceling the alike terms, one get[1]

$${\displaystyle a^{3}+b^{3}.}$$