SOLUTION: Factorise: 64x^3-(2y-1)^3

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Question 1094591: Factorise:
64x^3-(2y-1)^3
Found 2 solutions by ikleyn, Edwin McCravy:
Answer by ikleyn(52803) About Me  (Show Source):
You can put this solution on YOUR website!
.
Use the standard shortcut

a%5E3+-+b%5E3 = %28a-b%29%2A%28a%5E2+%2B+ab+%2B+b%5E2%29,


with a = 4x,  b = (2y-1).


1. The cube of the sum formula is                 %28a+%2B+b%29%5E3+=+a%5E3+%2B+3a%5E2%2Ab+%2B+3a%2Ab%5E2+%2B+b%5E3.
      For details and examples of applications of this formula see the lesson The cube of the sum formula in this site.


2. The cube of the difference formula is     %28a+-+b%29%5E3+=+a%5E3+-+3a%5E2%2Ab+%2B+3a%2Ab%5E2+-+b%5E3.
      For details and examples of applications of this formula see the lesson The cube of the difference formula in this site.


3. The sum of cubes formula is                       a%5E3+%2B+b%5E3+=+%28a+%2B+b%29%2A%28a%5E2+-+ab+%2B+b%5E2%29.
      For details and examples of applications of this formula see the lesson The sum of cubes formula in this site.

4. The difference of cubes formula is           a%5E3+-+b%5E3+=+%28a+-+b%29%2A%28a%5E2+%2B+ab+%2B+b%5E2%29.
      For details and examples of applications of this formula see the lesson The difference of cubes formula in this site.



Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
The factorization of

A%5E3-B%5E3 is

%28A%5E%22%22-B%29%28A%5E2%2BAB%2BB%5E2%29.

Since 64=4%2A4%2A4=4%5E3%29

64x%5E3-%282y-1%29%5E3

can be written as

%284x%29%5E3-%282y-1%29%5E3,

we can substitute A=(4x) and B=(2y-1) into

%28A%5E%22%22-B%29%28A%5E2%2BAB%2BB%5E2%29

%28%284x%29%5E%22%22-%282y-1%29%29%28%284x%29%5E2%2B%284x%29%282y-1%29%2B%282y-1%29%5E2%29

That simplifies to:

%284x%5E%22%22+-+2y+%2B+1%29+%2816x%5E2+%2B+8xy+-+4x+%2B+4y%5E2+-+4y+%2B+1%29

Edwin