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

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

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

           ~ Chúc bạn học tốt~

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

\(=x^3+3x^2yz+3xy^2z+3xyz^2+y^3+z^3-x^3-y^3-z^3\)

\(=3x^2yz+3xy^2z+3xyz^2\)

\(=3xyz\left(x+y+z\right)\)

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

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

\(=3\times\left(x+y\right)\times\left(y+z\right)\times\left(x+z\right)\)

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

\(=x^3-3x^2y+3xy^2-y^3+y^3-3y^2z+3yz^2-z^3+z^3-3z^2x+3zx^2-x^3\)

\(=-3x^2y+3xy^2-3y^2z+3yz^2-3z^2x+3zx^2\)

  =  -3xy(x-y) - 3yz(y-z) - 3zx(z-x)

Bạn có thể tham khảo tiếp bài của mình ở đây : https://olm.vn/hoi-dap/question/1264685.html

ta có : 

\(a^3+c^3=\left(a+c\right)^3-3ac\left(a+c\right)\)

nên \(a^3+c^3-b^3+3abc=\left(a+c\right)^3-b^3-3ac\left(a+c-b\right)\)

\(=\left(a+c-b\right)\left[\left(a+c\right)^2+b\left(a+c\right)+b^2-3ac\right]=\left(a+c-b\right)\left(a^2+b^2+c^2+ab+bc-ac\right)\)

b. tương tự ta có :

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

\(=\left(a-b-c\right)\left[a^2+a\left(b+c\right)+\left(b+c\right)^2-3bc\right]=\left(a-b-c\right)\left(a^2+b^2+c^2+ab+ac-bc\right)\)

c. ta có : \(\left(x-y\right)^3+\left(y-z\right)^3+\left(z-x\right)^3=\left(x-z+z-y\right)^3+\left(y-z\right)^3+\left(z-x\right)^3\)

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

\(=3\left(x-z\right)\left(z-y\right)\left(x-y\right)\)

bài a) bn trên đã dẫn link cho bn r

bài b)

Đặt x-y=a;y-z=b;z-x=c 

\(=>a+b+c=x-y+y-z+z-x=0\)

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

Theo câu a)\(a^3+b^3+c^3-3abc=\left(a+b+c\right)\left(a^2+b^2+c^2-ab-bc-ac\right)=0\) (do a+b+c=0)

\(=>a^3+b^3+c^3=3abc=>\left(x-y\right)^3+\left(y-z\right)^3+\left(z-x\right)^3=3\left(x-y\right)\left(y-z\right)\left(z-x\right)\)

a) Ta có :

\(a^3+b^3+c^3-3abc\)

\(\Rightarrow\left(a+b\right)^3-3ab\left(a+b\right)+c^3-3abc\)

\(\Rightarrow\left(a+b+c\right)\left[\left(a+b^2\right)-\left(a+b\right)c+c^2\right]-3ab\left(a+b+c\right)\)

\(\Rightarrow\left(a+b+c\right)\left(a^2+b^2+c^2-ab-bc-ca\right)\)

P/s tham khảo nha

hok tốt

\(a^3+b^3+c^3-3abc\)

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

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

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

Áp dụng \(\left(a+b\right)^3=a^3+b^3+3ab\left(a+b\right)\)

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

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

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

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

\(=3\left(x+y\right)\left(xy+xz+yz+z^2\right)\)

\(=3\left(x+y\right)\left[x\left(y+z\right)+z\left(y+z\right)\right]\)

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

a) \(x^4+5x^3+10x-4\)

\(=\left(x^4+2x^2\right)+\left(5x^3+10x\right)-\left(2x^2+4\right)\)

\(=x^2\left(x^2+2\right)+5x\left(x^2+2\right)-2\left(x^2+2\right)\)

\(=\left(x^2+2\right)\left(x^2+5x-2\right)\)

\(=\left(x^2+2\right)\left(x^2+2.x.\frac{5}{2}+\frac{25}{4}-\frac{25}{4}-2\right)\)

\(=\left(x^2+2\right)\left[\left(x+\frac{5}{2}\right)^2-\frac{33}{4}\right]\)

\(=\left(x^2+2\right)\left[\left(x+\frac{5}{2}\right)^2-\left(\frac{\sqrt{33}}{2}\right)^2\right]\)

\(=\left(x^2+2\right)\left(x+\frac{5}{2}-\frac{\sqrt{33}}{2}\right)\left(x^2+\frac{5}{2}+\frac{\sqrt{33}}{2}\right)\)

b) \(x^3+y^3+z^3-3xyz\)

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

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

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

\(=\left(x+y+z\right)\left(x^2+y^2+z^2-xy-zx-zy\right)\)

ai giúp hộ kìa