# Sum and Difference of Two Cubes

### From CATs

The difference of two cubes can be factored. That is, *A*^{3} − *B*^{3} can be factored. The factorization is given by

*A*^{3}−*B*^{3}= (*A*−*B*)(*A*^{2}+*A**B*+*B*^{2}).

We can check this factorization by multiplying out the right hand side and simplifying:

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*A*−*B*)(*A*^{2}+*A**B*+*B*^{2}) = (*A*−*B*)*A*^{2}+ (*A*−*B*)*A**B*+ (*A*−*B*)*B*^{2}=*A*^{3}−*A*^{2}*B*+*A*^{2}*B*−*A**B*^{2}+*A**B*^{2}−*B*^{3}=*A*^{3}−*B*^{3}.

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Unlike the sum of two squares (which factors only if we use complex numbers), the sum of two cubes can be factored. That is, *A*^{3} + *B*^{3} can be factored. The factorization is given by

*A*^{3}+*B*^{3}= (*A*+*B*)(*A*^{2}−*A**B*+*B*^{2}).

We can check this factorization by multiplying out the right hand side and simplifying:

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*A*+*B*)(*A*^{2}−*A**B*+*B*^{2}) = (*A*+*B*)*A*^{2}− (*A*+*B*)*A**B*+ (*A*+*B*)*B*^{2}=*A*^{3}+*A*^{2}*B*−*A*^{2}*B*−*A**B*^{2}+*A**B*^{2}+*B*^{3}=*A*^{3}+*B*^{3}.

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Let's look at a couple of examples.

Try a few yourself. Move your mouse over the solution box to see the solution. If you have trouble use the pencast below.

1. Factor *x*^{3}-27.

Solution

2. Factor 8*x*^{3}+*y*^{3}.

Solution