a) \(x\left(x^2-2x\right)+\left(x-2x\right)=x^2\left(x-2\right)+x\left(x-2\right)=\left(x-2\right)\left(x^2+x\right)⋮x-2\forall x,y\in Z\)

b) \(x^3y^2-3yx^2+xy=xy\left(x^2y-3x+1\right)⋮xy\forall x,y\in Z\)

c) \(x^3y^2-3x^2y^3+xy^2=xy^2\left(x^2-3xy+1\right)⋮\left(x^2-3xy+1\right)\forall x,y\in Z\)

A=x^3 + y^3 + 3xy(x+y)
  =x+3x^y+3xy^2+y^3
  =(x+y)^3=2^3=8
B=x^2+2xy+y^2+4
  =(x+y)^2+4=4+4=8

C=x^3+y^3+3xy(x+y)+7(x+y)

  =(x+y)^3+7(x+y)
  =2^3+7.2
  =8+14=22

Bài 2:

\(M=x^2-2xy+y^2=\left(x-y\right)^2=\left(-3\right)^2=9\)

\(N=x^2+y^2=\left(x-y\right)^2+2xy=9+2.10=29\)

\(P=x^3-3x^2y+3xy^2-y^3=\left(x-y\right)^3=\left(-3\right)^3=-27\)

\(Q=x^3-y^3=\left(x-y\right)^3+3xy\left(x-y\right)=\left(-3\right)^3+3.10.\left(-3\right)=-117\)

Bài 1:

a)  \(A=x^2+2xy+y^2=\left(x+y\right)^2=\left(-1\right)^2=1\)

b)  \(B=x^2+y^2=\left(x+y\right)^2-2xy=\left(-1\right)^2-2.\left(-12\right)=25\)

c)  \(C=x^3+3x^2y+3xy^2+y^3=\left(x+y\right)^3=\left(-1\right)^3=-1\)

d)  \(D=x^3+y^3=\left(x+y\right)^3-3xy\left(x+y\right)=\left(-1\right)^3-3.\left(-12\right).\left(-1\right)=-37\)

a) Ta có: A = x³ + y³ + 3xy = (x + y)(x² - xy + y²) + 3xy = 1. (x² - xy + y²) + 3xy = x² - xy + y² + 3xy = x² + 2xy + y² = (x + y)² = 1² = 1

b)Ta có: B = x³ - y³ - 3xy = (x - y)(x² + xy + y²) - 3xy = 1. (x² + xy + y²) - 3xy = x² + xy + y² - 3xy = x² - 2xy + y² = (x - y)² = 1² = 1

d) Ta có : D = x³ + y³ + 3xy(x² + y²) + 6x²y²(x + y)

=> D = (x + y)(x² - xy + y²) + 3xy(x² + 2xy + y²) -  6x²y² + 6x²y²

=> D = x² - xy + y² + 3xy(x + y)² 

=> D = x² - xy + y² + 3xy.1²

=> D = x² + 2xy + y²

=> D = (x + y)² = 1² = 1

a) \(A=x^3+y^3+3xy\)

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

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

\(=\left(x+y\right)^2=1^2=1\)

b) \(B=x^3-y^3-3xy\)

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

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

\(=\left(x-y\right)^2=1^2=1\)

c) Ta có: \(x^2+y^2=10\)

\(\Leftrightarrow\left(x+y\right)^2-2xy=10\)

\(\Leftrightarrow4-2xy=10\Leftrightarrow2xy=-6\Leftrightarrow xy=-3\)

 \(C=x^3+y^3\)

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

\(=2\left(10+3\right)=26\)

a. Có \(x+y=2\Rightarrow x^2+2xy+y^2=4\Rightarrow x^2+y^2=4-2.\left(-3\right)=10\)

\(x^4+y^4=\left(x^2\right)^2+\left(y^2\right)^2=\left(x^2+y^2\right)^2-2x^2y^2\)

\(=10^2-2.\left(-3\right)^2=82\)

b. Ta có \(x+y=1\Rightarrow x^2+y^2=1-2xy\)

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

\(=1.\left(1-2xy-xy\right)+3xy=1\)

Các câu còn lại tương tự

\(A=x^3+y^3+3xy=\left(x+y\right)^3-3xy\left(x+y\right)+3xy=1+0=1\)

\(B=\left(x-y\right)^3+3xy\left(x-y\right)-3xy=1\)

\(c,M=a^2-ab+b^2+3ab\left(a^2+b^2\right)+6a^2b^2=3ab\left(a^2+2ab+b^2\right)+a^2-ab+b^2\)

\(=3ab+a^2-ab+b^2=\left(a+b\right)^2=1\)

\(x+y=2;x^2+y^2=10\text{ do đó:}xy=-3\text{ nên }\left(x-y\right)^2=16\text{ do đó: }x-y=4\text{ hoặc }x-y=-4\)

\(\text{giải ra được:}x=3;y=-1\text{ hoặc ngược lại nên }x^3+y^3=-26\text{ hoặc }26\)

A = x³ + y³ + 3xy

= x³ + 3x²y + 3xy² + y³ - 3x²y - 3xy² + 3xy

= ( x³ + 3x² + 3xy² + y³ ) - ( 3x²y + 3xy - 3xy )

= ( x + y )³ - 3xy( x + y - 1 )

= 1³ - 3xy( 1 - 1 )

= 1³ - 3xy.0

= 1 - 0 = 1

Vậy A = 1

b) B = x³ - y³ - 3xy

= x³ - 3x²y + 3xy² - y³ + 3x²y - 3xy² - 3xy

= ( x³ - 3x²y + 3xy² - y³ ) + ( 3x²y - 3xy² - 3xy )

= ( x - y )³ + 3xy( x - y - 1 )

= 1³ + 3xy( 1 - 1 )

= 1 + 3xy.0

= 1 + 0 = 1

Vậy B = 1

M = a³ + b³ + 3ab( a² + b² ) + 6a²b²( a + b )

= ( a + b )( a² - ab + b² ) + 3ab[ ( a + b )² - 2ab ] + 6a²b²( a + b )

= ( a + b )[ ( a + b )² - 3ab ] + 3ab[ ( a + b )² - 2ab ] + 6a²b²( a + b )

= 1.( 1 - 3ab ) + 3ab( 1 - 2ab ) + 6a²b².1

= 1 - 3ab + 3ab - 6a²b² + 6a²b²

= 1

Vậy M = 1

d) x + y = 2

⇔ ( x + y )² = 4

⇔ x² + 2xy + y² = 4

⇔ 10 + 2xy = 4 ( gt x² + y² = 10 )

⇔ 2xy = -6

⇔ xy = -3

x³ + y³ = x³ + 3x²y + 3xy² + y³ - 3x²y - 3xy²

            = ( x³ + 3x²y + 3xy² + y³ ) - ( 3x²y + 3xy² )

            = ( x + y )³ - 3xy( x + y )

            = 2³ - 3.(-3).(2)

            = 8 + 18 = 26

d đâu bạn

Các bài này đưa về dạng Hằng đẳng thức là được . Làm ra dài lắm bạn ạ !

Ns nghe dễ lắm, lm thử đee =))

A. \(x+y=1\Rightarrow\left(x+y\right)^3=1\Rightarrow x^3+3x^2y+3xy^2+y^3=1\)

\(\Rightarrow x^3+3xy\cdot\left(x+y\right)+y^3=1\)

\(\Rightarrow x^3+3xy+y^3=1\)

B. \(x-y=1\Rightarrow\left(x-y\right)^3=1\Rightarrow x^3-3x^2y+3xy^2-y^3=1\)

\(\Rightarrow x^3-3xy\cdot\left(x-y\right)-y^3=1\)

\(\Rightarrow x^3-3xy-y^3=1\)

C. \(M=a^3+b^3+3ab\left[\left(a+b\right)^2-2ab\right]+6a^2b^2\left(a+b\right)\)

\(M=a^3+b^3+3ab\left(1-2ab\right)+6a^2b^2=a^3+b^3+3ab-6a^2b^2+6a^2b^2\)

\(M=a^3+b^3+3ab=1\)(Theo hệ quả câu A).

D. Từ \(x+y=2\Rightarrow\left(x+y\right)^2=4\Rightarrow x^2+y^2+2xy=4\Rightarrow10+2xy=4\Rightarrow xy=-3\)

Mà, \(\left(x+y\right)^3=x^3+y^3+3xy\left(x+y\right)\)

\(\Leftrightarrow2^3=x^3+y^3+3\left(-2\right)\cdot2\Leftrightarrow x^3+y^3=8+12=20\)