a,\(\left(a+b+c\right)^3=a^3+b^3+c^3+3\left(a^2b+b^2a+c^2a+ca^2+b^2c+c^2b\right)\)

Tương tự :

\(\left(b+c-a\right)^3=b^3+c^3-a^3+3\left(a^2b-b^2a+ca^2-ac^2+b^2c+c^2b\right)\)

\(\left(b+a-c\right)^3=b^3-c^3+a^3+3\left(a^2b+b^2a-ca^2+ac^2-b^2c+c^2b\right)\)

\(\left(a+c-b\right)^3=c^3+a^3-b^3+3\left(-a^2b+b^2a+ca^2+ac^2+b^2c-c^2b\right)\)

Biểu thức sau khi rút gọn ta được 

24abc

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

\(\left(c+b\right)^3=c^3+b^3+3\left(c^2b+b^2c\right)\)

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

=>\(\left(a+b\right)^3+\left(b+c\right)^3+\left(c+a\right)^3=\)\(2\left(a^2+b^2+c^2\right)+3\left(a^2b+b^2a+c^2a+ca^2+b^2c+c^2b\right)\)

Lại có 

\(3\left(a+b\right)\left(b+c\right)\left(c+a\right)=\left(3\left(a^2b+b^2a+c^2a+ca^2+b^2c+c^2b+2abc\right)\right)\)

Biểu thức khi đó trở thành 

\(2\left(a^2+b^2+c^2\right)-6abc=2\left(a^2+b^2+c^2-3abc\right)\)

Tặng vk iu 

Đặt x = a+b , y = b+c , z = c+a

Khi đó : \(\left(a+b\right)^3+\left(b+c\right)^3+\left(c+a\right)^3-3\left(a+b\right)\left(b+c\right)\left(c+a\right)\)

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

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

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

\(=\frac{x+y+z}{2}\left[\left(x-y\right)^2+\left(y-z\right)^2+\left(z-x\right)^2\right]\)

Thay x,y,z bởi a,b,c vào và rút gọn :)

Đặt\(a+b=x\)

\(b+c=y\)

\(c+a=z\)

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

\(=\frac{1}{2}\left(x+y+z\right)\left[\left(x-y\right)^2+\left(y-z\right)^2+\left(x-z\right)^2\right]\)

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

Sửa đề cho nó đẹp

\(\frac{\left(a-b\right)^3+\left(b-c\right)^3+\left(c-a\right)^3}{a^2\left(b-c\right)+b^2\left(c-a\right)+c^2\left(a-b\right)}\)

\(=\frac{3\left(a-b\right)\left(a-c\right)\left(c-b\right)}{\left(a-b\right)\left(a-c\right)\left(b-c\right)}=-3\)

em ms hok lớp 1

(a+b)³+(b+c)³ +(c+a)³-3(a+b)(b+c)(c+a)

=(a+b+b+c+c+a)[ (a+b)²+(b+c)²+(c+a)²-(a+b)(b+c)-(b+c)(c+a)-(c+a)(a+b)]

=2(a+b+c)[(a+b)²+(b+c)²+(c+a)² -(a+b)(b+c)-(b+c)(c+a)-(c+a)(a+b)]

Áp dụng hđt a³+b³+c³=3abc ta có

(a+b)³+(b+c)³+(c+a)³=3(a+b)(b+c)(c+a)

→đề bài có giá trị =0

(a+b)³+(b+c)³+(c+a)³-3(a+b)(b+c)(c+a)

bạn phân tích ra theo HĐT và nhân đôn thức vs đa thức sẽ dc

=2(a³+b³+c³-3abc)

\(\frac{a^3}{\left(a-b\right)\left(a-c\right)}+\frac{b^3}{\left(b-c\right)\left(b-a\right)}+\frac{c^3}{\left(c-a\right)\left(c-b\right)}\)

\(=\frac{a^3\left(b-c\right)}{\left(a-b\right)\left(a-c\right)\left(b-c\right)}+\frac{b^3\left(c-a\right)}{\left(b-c\right)\left(a-b\right)\left(a-c\right)}+\frac{c^3\left(a-b\right)}{\left(a-c\right)\left(b-c\right)\left(a-b\right)}\)

\(=\frac{a^3\left(b-c\right)+b^3\left(c-a\right)+c^3\left(a-b\right)}{\left(a-b\right)\left(a-c\right)\left(b-c\right)}\)

\(=\frac{\left(a-b\right)\left(a-c\right)\left(b-c\right)\left(a+b+c\right)}{\left(a-b\right)\left(a-c\right)\left(b-c\right)}=a+b+c\)

Phân tích mẫu thức thành nhân tử :

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

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

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

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

\(=\left(b-c\right)\left[a\left(a-b\right)-c\left(a-b\right)\right]=\left(b-c\right)\left(a-c\right)\left(a-b\right).\)

Do đó : \(A=\frac{\left(b-c\right)^3+\left(c-a\right)^3+\left(a-b\right)^3}{-\left(a-b\right)\left(b-c\right)\left(c-a\right)}\)

Nhận xét : Nếu \(x+y+z=0\) thì \(x^3+y^3+z^3=3xyz.\)

Đặt \(b-c=x,c-a=y,a-b=z\) thì \(x+y+z=0\)

Theo nhận xét trên : \(A=\frac{x^3+y^3+z^3}{-xyz}=\frac{3xyz}{-xyz}=-3.\)

Tử:

(b - c)³ + (c - a)³ + (a - b)³

= (b - c + c - a + a - b)³ - 3(b - c + c - a)(b - c + a - b)(c - a + a - b)

= 0 - 3(b - a)(a - c)(c - b)

= 3(a - b)(a - c)(c - b)

Mẫu:

a²(b - c) + b²(c - a) + c²(a - b)

= a²(b - c) + b²c - ab² + ac² - bc²

= a²(b - c) - a(b² - c²) + bc(b - c)

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

= (b - c)(a² - ab - ac + bc)

= (b - c)[a(a - b) - c(a - b)]

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

\(A=\frac{3\left(a-b\right)\left(a-c\right)\left(c-b\right)}{\left(b-c\right)\left(a-b\right)\left(a-c\right)}\)

\(=\frac{3\left(c-b\right)}{b-c}\)