How do you factor $(a+b)³ - (a-b)³$ ?

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Answer 1 · 2015-05-23T23:50:06

Expanding:

$(a+b)^3-(a-b)^3$

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

$= 6a^2b+2b^3$

$=2b(3a^2+b^2)$

If you are allowed complex coefficients this can be broken down into linear factors:

$=2b(sqrt(3)a+ib)(sqrt(3)a-ib)$

Notice also that:

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

How do you factor (a+b)³ - (a-b)³(a+b)³ - (a-b)³?