Question

phân tích đa thức thành nhân tử: a(b^2-c^2)-b(a^2-c^2)+c(a^2-b^2)

Answer 1

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

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

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

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

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

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

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

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

Answer 2

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

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

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

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

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

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