Sum of Two Fifth Powers

Theorem

$x^5 + y^5 = \paren {x + y} \paren {x^4 - x^3 y + x^2 y^2 - x y^3 + y^4}$


Proof

From Sum of Two Odd Powers:

$a^{2 n + 1} + b^{2 n + 1} = \paren {a + b} \paren {a^{2 n} - a^{2 n - 1} b + a^{2 n - 2} b^2 - \dotsb + a b^{2 n - 1} + b^{2 n} }$

We have that $5 = 2 \times 2 + 1$.

The result follows directly by setting $n = 2$.

$\blacksquare$


Sources

  • 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 2$: Special Products and Factors: $2.16$
  • 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 2$: Special Products and Factors: $2.16.$
This article is issued from ProofWiki. The text is available under Creative Commons Attribution-ShareAlike License unless otherwise noted. Additional terms may apply for the media files.