Question

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

Answer 1

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

Answer 2

Đặt biểu thuucs là A suy ra

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

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

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

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

=(b-c)(b+c)(a+b)+(a-b)(a+b)(b+c)

=(b+c)(a+b)(b-c+a-b)=(b+c)(a+b)(a-c)