Question

a(b^2-c^2)+b(c^2-a^2)+c(a^2-b^2)

Answer 1

a(b - c)(b + c) - bc(b - c) - a² (b - c) = (b - c)(ab + ac - bc - a²)

Answer 2

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

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

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

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

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

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

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

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