Question

Phân tích thành nhân tử :

a) \(A=\left(x^2+y^2\right)^3+\left(z^2-x^2\right)^3-\left(y^2+z^2\right)^3\)

b) \(B=\left(x+y+z\right)^3-x^3-y^3-z^3\)

 

Answer 1

Đặt \(x^2+y^2=a;y^2+z^2=b\)

\(\Rightarrow z^2-x^2=\left(y^2+z^2\right)-\left(x^2+y^2\right)=b-a\)

\(\Rightarrow A=a^3+\left(b-a\right)^3-b^3\)

\(=\left(a-b\right)\left(a^2+ab+b^2\right)-\left(a-b\right)^3\)

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

\(=3ab\left(a-b\right)=3\left(x^2+y^2\right)\left(y^2+z^2\right)\left(x^2-z^2\right)\)

\(=3\left(x^2+y^2\right)\left(y^2+z^2\right)\left(x-z\right)\left(x+z\right)\)

Answer 2

\(B=\left(x+y+z\right)^3-x^3-y^3-z^3\)

\(B=x^3+y^3+z^3+3.\left(x+y\right)\left(y+z\right)\left(z+x\right)-x^3-y^3-z^3\)

\(B=3.\left(x+y\right)\left(y+z\right)\left(z+x\right)\)

Đây là hằng đẳng thức:

\(\left(a+b+c\right)^3=a^3+b^3+c^3+3\left(a+b\right)\left(b+c\right)\left(c+a\right)\)