How do you factor $(2x+1)^3 + (2y)^3$ ?

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Answer 1 · 2015-05-28T19:46:23

For any numbers a and b, $a^3 + b^3 = (a+b)(a^2-ab+b^2)$

So, this expression becomes:

$((2x+1)+2y)((2x+1)^2-(2x+1)(2y)+(2y)^2)$

$=(2x+2y+1)((4x^2+4x+1)-(4xy+2y)+(4y^2))$

$=(2x+2y+1)(4x^2+4y^2-4xy+4x-2y+1)$

These 2 numbers look like they can be factored, but they can't because x and y are different variables.

Answer 2 · 2015-05-28T19:48:13

You can use the identity for a sum of cubes:

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

substituting $a=2x+1$ and $b=2y$ to get:

$(2x+1)^3+(2y)^3$

$= ((2x+1)+2y)((2x+1)^2 - (2x+1)2y + (2y)^2)$

$= (2x+2y+1)((4x^2+4x+1)-(4xy+2y)+4y^2)$

$= (2x+2y+1)(4x^2-4xy+4y^2+4x-2y+1)$

How do you factor (2x+1)^3 + (2y)^3(2x+1)^3 + (2y)^3?