SOLUTION: factors this with the product of the sum and difference of 2 cubes (x^(2m)-1)^3-x^(6m)

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Question 1086329: factors this with the product of the sum and difference of 2 cubes (x^(2m)-1)^3-x^(6m)
Answer by Edwin McCravy(20064) About Me  (Show Source):
You can put this solution on YOUR website!
%28x%5E%282m%29-1%29%5E3-x%5E%286m%29

Write the 2nd term, x%5E%286m%29, as the cube %28x%5E%282m%29%29%5E3

%28x%5E%282m%29-1%29%5E3-%28x%5E%282m%29%29%5E3

Use the factorization principle

A%5E3-B%5E3%22%22=%22%22%28A-B%29%28A%5E2%2BAB%2BC%29

Substitute %28x%5E%282m%29-1%29 for A and %28x%5E%282m%29%29 for B 

%28x%5E%282m%29-1%29%5E3%5E%22%22-%28x%5E%282m%29%29%5E3%22%22=%22%22

Remove the inner parentheses on the right, all but %28x%5E%282m%29-1%29%5E2.


%28x%5E%282m%29-1%29%5E3%5E%22%22-%28x%5E%282m%29%29%5E3%22%22=%22%22


Combine like terms:


%28x%5E%282m%29-1%29%5E3%5E%22%22-%28x%5E%282m%29%29%5E3%22%22=%22%22%28-1%5E%22%22%29%28%28x%5E%282m%29-1%29%5E2%2B2x%5E%284m%29-x%5E%282m%29%29

Square out %28x%5E%282m%29-1%29%5E2 as %28x%5E%282m%29-1%29%28x%5E%282m%29-1%29=+x%5E%284m%29-2x%5E%282m%29%2B1,
and substitute:


%28x%5E%282m%29-1%29%5E3%5E%22%22-%28x%5E%282m%29%29%5E3%22%22=%22%22%28-1%5E%22%22%29%28x%5E%284m%29-2x%5E%282m%29%2B1%2B2x%5E%284m%29-x%5E%282m%29%29

%28x%5E%282m%29-1%29%5E3%5E%22%22-%28x%5E%282m%29%29%5E3%22%22=%22%22%28-1%5E%22%22%29%283x%5E%284m%29-3x%5E%282m%29%2B1%29


%28x%5E%282m%29-1%29%5E3%5E%22%22-%28x%5E%282m%29%29%5E3%22%22=%22%22-3x%5E%284m%29%2B3x%5E%282m%29-1

Edwin