Problem
ALG-B1-M02-P023 Factoring by Vanishing
#23
★★★★☆ Level 4 of 5
Prove that \(a^2(b-c)+b^2(c-a)+c^2(a-b)=-(a-b)(b-c)(c-a)\).
Both sides have degree \(3\). Check the factors \(a-b\), \(b-c\), \(c-a\), then determine the sign.
The left side vanishes when \(a=b\), when \(b=c\), and when \(c=a\). Hence it is divisible by \((a-b)(b-c)(c-a)\).
Since the degree is \(3\), the left side equals \(k(a-b)(b-c)(c-a)\). Substitute \(a=1\), \(b=2\), \(c=3\). The left side is \(-2\), while the product is \(2\). Thus \(k=-1\).
The identity is proved.
This is an important transition from mechanical to structural factorisation.