Problem
ALG-B1-M02-P019 A Cyclic Expression
#19
★★★☆☆ Level 3 of 5
Factor \(a(b^2-c^2)+b(c^2-a^2)+c(a^2-b^2)\).
Check that the expression vanishes when \(a=b\), \(b=c\), and \(c=a\).
Let the expression be \(E\). If \(a=b\), then \(E=0\), so \(a-b\) is a factor. Similarly, \(b-c\) and \(c-a\) are factors.
Since the degree is \(3\), \(E=k(a-b)(b-c)(c-a)\). Substitute \(a=1\), \(b=2\), \(c=3\): \(E=2\), and \((a-b)(b-c)(c-a)=2\). Hence \(k=1\).
Thus \(E=(a-b)(b-c)(c-a)\).
This is olympiad-style factor detection through vanishing.