Problem
ALG-B1-M02-P016 A Divisor of a Sixth Power Difference
#16
★★★☆☆ Level 3 of 5
Prove that \(a^6-b^6\) is divisible by \(a^2+ab+b^2\).
Factor \(a^6-b^6\) as \((a^3-b^3)(a^3+b^3)\).
\(a^6-b^6=(a^3-b^3)(a^3+b^3)\).
Also \(a^3-b^3=(a-b)(a^2+ab+b^2)\). Therefore \(a^2+ab+b^2\) is indeed a factor of \(a^6-b^6\).
The problem shows that the needed factor may appear in the second factorisation step.