How do you factor $(a+2b)^3 - (a-2b)^3$ ?

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Answer 1 · 2015-04-15T15:43:06

Remembering that a difference of cubes can be factored:

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

then:

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

$=[(a+2b) - (a-2b)][(a+2b)^2+(a+2b)(a-2b)+(a-2b)^2]=$

$=(a+2b-a+2b)(a^2+4ab+4b^2+a^2-4b^2+a^2-4ab+4b^2]=$

$=4b(3a^2+4b^2)$ .

But, in this case, there is a faster and easier way:

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

$=a^3+6a^2b+12ab^2+8b^3-a^3+6a^2b-12ab^2+8b^3=$

$=12a^2b+16b^3=4b(3a^2+4b^2)$ .

How do you factor (a+2b)^3 - (a-2b)^3(a+2b)^3 - (a-2b)^3?