Question

Phân tích bằng phương pháp xét giá trị riêng:

a(b²-c²)+b(c²-a²)+c(a²-b²)

Answer 1

\(a\left(b^2-c^2\right)+b\left(c^2-a^2\right)+c\left(a^2-b^2\right)\) \(=ab^2-ac^2+bc^2-ba^2+ca^2-cb^2\)

\(=\left(ab^2-cb^2\right)+\left(ca^2-ac^2\right)+\left(bc^2-ba^2\right)\)

\(=b^2\left(a-c\right)+ac\left(a-c\right)-b\left(a-c\right)\left(a+c\right)\)

\(=\left(a-c\right)\left(b^2+ac-ab-bc\right)\)

\(=\left(a-c\right)\left(\left(b^2-bc\right)-\left(ab-ac\right)\right)\)

\(=\left(a-b\right)\left(b-c\right)\left(c-a\right)\)