8.5 - Factoring a sum or difference of cubes


Just as the names suggest, a sum of cubes is an expression of the form:
a 3 + b 3,
and a difference of cubes is an expression of the form:
a 3b 3.
A sum of cubes can be factored like this:
a 3 + b 3 = (a + b) (a 2 a b + b 2 ),
and a difference of cubes can be factored like this:
a 3 b 3 = (a b) (a 2 + a b + b 2 ).
Notice the signs shown in red. The binomial on the right has the same sign as the binomial on the left and the trinomial on the right has the opposite sign.


You can easily verify both these factors by multiplying out the right hand sides and noticing that all the terms except the cube terms cancel. Here are some examples. In each case we recognize the form a 3 ± b 3, (± is an abbreviation for “plus or minus”), then identify a and b, and then simply state the factored form.




Example: Factor   x 3 + 27.


Example: Factor   8 x 6 − 64 y 3.


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