Algebra 1 The Difference of Two Cubes
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A Bigger Difference
The Difference of Two Cubes

 

 So when we have something that is made up of two different terms

 that have been multiplied times themselves three times,
 we have the difference of two cubes.
 
 So X 3 - 1 factors like this ...
 

X3 - 1 = (X - 1)(X2 + X + 1)

 
 In general, if you have a 3X 3 - b 3, it factors to
 

(a3X3 - b3) = (a - b)(a2X2 + abX + b2)

 
 Examples:
X3 - 8 =    (X - 2)(X2 + 2X + 4)  8 = 23
27X3 - 1 =   (3X - 1)(9X2 + 3X + 1) 27 = 33
 
 Like with the squares, the numbers 
 don't have to be cubes of integers.
 
 We can even write X - 1 
 as the difference of two cubes.
 

 
 We couldn't factor X 2 + 1, can we factor X 3 + 1?
 
 Yes we can!!!
 
 It works out almost like the difference of two cubes.
 
 The only changes are that the sign in the first factor is positive
 and the middle sign in the second factor is a negative.
 So:

X3 + 1 = (X + 1)(X2 - X + 1)

 
 Using our friends "a" and "b." 
 We can say ...
 

(a3X3 + b3) = (a2X2 - abX + b2)

 
 This one is called factoring the sum of two cubes.
 
 Study these two different factoring plans for a moment or two.
 The only difference is where the negative sign goes ...
 

(a3X3 - b3) = (a2X2 + abX + b2)

(a3X3 + b3) = (a2X2 - abX + b2)

 

   copyright 2005 Bruce Kirkpatrick

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