Difference of Two Powers/Examples/Difference of Two Cubes

Theorem

$x^3 - y^3 = \paren {x - y} \paren {x^2 + x y + y^2}$

Corollary

$x^3 - 1 = \paren {x - 1} \paren {x^2 + x + 1}$

Proof

From Difference of Two Powers:

$\ds a^n - b^n = \paren {a - b} \sum_{j \mathop = 0}^{n - 1} a^{n - j - 1} b^j$

The result follows directly by setting $n = 3$.

$\blacksquare$

Sources

• 1946: Alfred Tarski: Introduction to Logic and to the Methodology of Deductive Sciences (2nd ed.) ... (previous) ... (next): $\S 1.5$: The Importance of Variables in Mathematics
• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 2$: Special Products and Factors: $2.12$