Basic Technical Mathematics, 11th Edition

6.3 The Sum and Difference of Cubes

• Factoring the Sum of Cubes • Factoring the Difference of Cubes • Summary of Methods of Factoring

We have seen that the difference of squares can be factored, but that the sum of squares often cannot be factored. We now turn our attention to the sum and difference of cubes, both of which can be factored. The following two equations show us how either a sum or difference of cubes can be factored into the product of a binomial and a trinomial. (The proof is left as an exercise.)

$$a^3\hspace{0.17em}+\hspace{0.17em}{b}^3\hspace{0.17em}=\hspace{0.17em}(a\hspace{0.17em}+\hspace{0.17em}b)(a^2\hspace{0.17em}-\hspace{0.17em}ab\hspace{0.17em}+\hspace{0.17em}{b}^2)$$

(6.8)

$$a^3\hspace{0.17em}-\hspace{0.17em}{b}^3\hspace{0.17em}=\hspace{0.17em}(a\hspace{0.17em}-\hspace{0.17em}b)(a^2\hspace{0.17em}+\hspace{0.17em}ab\hspace{0.17em}+\hspace{0.17em}{b}^2)$$

(6.9)

Note carefully the positions of the $\hspace{0.17em}+\hspace{0.17em}$ and $\hspace{0.17em}-\hspace{0.17em}$ signs.

In these equations, the second factors are usually prime.

EXAMPLE 1 Factoring sum and difference of ...