Binomial Theorem/Examples/Cube of Difference

Example of Use of Binomial Theorem

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

Proof

Follows directly from the Binomial Theorem:

$\ds \forall n \in \Z_{\ge 0}: \paren {x + \paren {-y} }^n = \sum_{k \mathop = 0}^n \binom n k x^{n - k} \paren {-y}^k$

putting $n = 3$.

$\blacksquare$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 2$: Special Products and Factors: $2.4$
• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 20$: Binomial Series: $20.6$