8.5 - Factoring a sum or difference of cubes

Just as the names suggest, a sum of cubes is an expression of the form:

a³ + b³,

and a difference of cubes is an expression of the form:

a³ − b³.

A sum of cubes can be factored like this:

a³ + b³ = (a + b) (a² − a b + b² ),

and a difference of cubes can be factored like this:

a³ − b³ = (a − b) (a² + a b + b² ).

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 a³ ± b³, (± is an abbreviation for “plus or minus”), then identify a and b, and then simply state the factored form.

Example: Factor x³ + 27.

Think of 27 as 3³. Then this is a sum of cubes and we can apply the sum of cubes formula:
a³ + b³ = (a + b) (a² − a b + b² ).
Substituting a = x and b = 3 into the formula yields:
x³ + 27 = (x + 3) (x² − 3 x + 9).

Example: Factor 8 x⁶ − 64 y³.

Think of 8 x⁶ as (2 x²)³ and 64 y³ as (4 y)³. Then this is a difference of cubes and we can apply the difference of cubes formula:
a³ − b³ = (a − b) (a² + a b + b² ).
Substituting a = 2 x² and b = 4 y into the formula yields:
8 x⁶ − 64 y³ = (2 x² − 4 y) (4 x⁴ + 8 x² y + 16 y² ).

Algebra Coach Exercises

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