a. ( a + b + c)² + a² + b² + c²

= a² + b² + c² + 2ab + 2ac + 2bc + a² + b² + c²

= (a+b)² + (b+c)² + (a+c)²

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

đặt a - b = x; b-c = y; c-a = z => x + y + z = 0 (1)

ta có: 2.x.(-y) + 2.(-x).z + 2.y.(-z)

= -2xy - 2xz - 2yz  = -2.(xy+xz+yz)  

ta có: (x+y+z)² = x² + y² + z² + 2xy + 2yz + 2xz

 0² = x² + y² + z² + 2.(xy+yz+xz)

=> x² + y² + z²  = -2.(xy+yz+xz) (2)

Từ (2) => 2.(a-b).(c-b) + 2.(b-a) .(c-a) + 2.(b-c).(a-c) = x² + y² + z²

= (a-b)² + (b-c)² + (c-a)²

Bài 2 :

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

\(A=a^2+b^2+c^2+2ab+2ac+2bc+a^2+b^2+c^2\)

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

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

a: \(=a^2-b^4\)

b: \(=\left(a^2+2a\right)^2-9\)

c: \(=a^2-\left(2a+3\right)^2\)

d: \(=a^4-\left(2a-3\right)^2\)

e: \(=\left(-a^2-2a+3\right)^2\)

g: \(=4a^2-a^4\)

Bài cuối hơi khó nhìn, bạn thông cảm nhé! ^^

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\(=\left(a-c\right)\left(ba^2+abc+bc^2-a^2c-ac^2-b^3\right)\)

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

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

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

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

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

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

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

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

Khúc cuối nè

1a) a² + b² + c² + 2ab + 2bc + 2ca + a² + b² + c²

= ( a² + 2ab +b² ) + ( a² + 2ac + c² ) + ( b² + 2bc + c² )

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

1b) 2.( ac - ab - bc + b² ) + 2.( bc - ba - ac + a² ) + 2.( ba - bc - ca + c² )

= 2ac - 2ab - 2bc + 2b² + 2bc - 2ab - 2ac +2a² + 2ab - 2bc - 2ac + 2c²

= 2a² + 2b² + 2c² - 2ab - 2ac - 2bc

= ( a² - 2ab + b² ) + (a² - 2ac + c² ) + (b² - 2bc + c² )

= (a-b)² + (a-c)² + (b-c)²

nhiều thế, đăng ít một thôi bạn

a/ \(A=\left(3+1\right)\left(3^2+1\right)\left(3^4+1\right)...\left(3^{64}+1\right)\)

\(2A=2\left(3+1\right)\left(3^2+1\right)\left(3^4+1\right)...\left(3^{64}+1\right)\)

\(2A=\left(3-1\right)\left(3+1\right)\left(3^2+1\right)\left(3^4+1\right)...\left(3^{64}+1\right)\)

\(2A=\left(3^2-1\right)\left(3^2+1\right)\left(3^4+1\right)...\left(3^{64}+1\right)\)

\(2A=\left(3^4-1\right)\left(3^4+1\right)...\left(3^{64}+1\right)\)

\(\Rightarrow2A=3^{128}-1\Rightarrow A=\dfrac{3^{128}-1}{2}\)

e) ta dể dàng thấy được : \(a^2+b^2=\left(a+b\right)^2-2ab\)

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

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

\(=4\left(a+b\right)^2-2\left(\left(a+b\right)^2-c^2\right)-2\left(a+b\right)^2\)

\(=4\left(a+b\right)^2-2\left(a+b\right)^2+2c^2-2\left(a+b\right)^2=2c^2\)

g) củng sử dụng cái trên ta có : \(G=\left(a+b+c+d\right)^2+\left(a+b-c-d\right)^2+\left(a+c-b-d\right)^2+\left(a+d-b-c\right)^2\)

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

\(=4\left(a+b\right)^2+4\left(a-b\right)^2-2\left(\left(a+b\right)^2-\left(c+d\right)^2\right)-2\left(\left(a-b\right)^2-\left(c-d\right)^2\right)\)

\(=4\left(\left(a+b\right)^2+\left(a-b\right)^2\right)-2\left(\left(a+b\right)^2+\left(a-b\right)^2\right)+2\left(\left(c+d\right)^2+\left(c-d\right)^2\right)\)

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

\(=2\left(4a^2-2\left(a^2-b^2\right)\right)+2\left(4c^2-2\left(c^2-d^2\right)\right)\)

\(=2\left(2a^2+2b^2\right)+2\left(2c^2+2d^2\right)=4\left(a^2+b^2+c^2+d^2\right)\)

bn đăng nhiều quá nên mk làm câu nào hay câu đó nha

mà nè mấy câu a;b;c;d hình như trên mạng có bn lên đó tìm nha .