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x^3-8 How to change this to (x-2)(x^2+2x+4) ?

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2 Answers

This is the difference of two cubes.  It is similar to a difference of two squares in that there is a formula for factoring it.
 
This is the formula
(a3 - b3) = (a2 +ab + b2)
In this case a = x since (x)3 = xand b = 2, since 23 = 8.
 
So plug in x for a, and 2 for b into the following:
(a3 - b3) = (a2 +ab + b2)
 
 
Giving you:
(x3 - 8) = (x-2)(x2 + (x)(2) + 22) = (x-2)(x2 + 2x + 4)
 
Check
 
x(x2 +2x + 4) - 2(x2 + 2x + 4)
 
x3 +2x2 +4x -2x2 -4x - 8
 
The x2 terms and the x terms cancel out and you are left with:
 
x3 - 8
 
Note that if it had been x3 + 8, the factors would be (x+2)(x2-2x+4)

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