Problem
ALG-B1-M01-P011 Difference of Powers
#11
★★☆☆☆ Level 2 of 5
Prove that for every positive integer \(n\), the expression \(a^n-b^n\) is divisible by \(a-b\).
Use \(a^n-b^n=(a-b)(a^{n-1}+a^{n-2}b+\cdots+b^{n-1})\).
Write \[ a^n-b^n=(a-b)(a^{n-1}+a^{n-2}b+\cdots+ab^{n-2}+b^{n-1}). \] Expanding shows that all intermediate terms cancel.
Hence \(a^n-b^n\) has factor \(a-b\).
This is an important bridge to the factorisation module.