(x+y)^2  =a^2

x^2 +2xy +y^2 =a^2

x^2+y^2 =a^2-2xy =a^2 -2b

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

             =a(a^2-2b-b)

            =a(a^2-3b)

            =a^3- 3ab

(x^2 +y^2)^2=(a^2-2b)^2  ( cái này tính cho x^4 + y^4)

tương tự như câu đầu tiên 

x^5+ y^5 (cái đó mình không biết)

sai con khi

\(1.\)

\(a)\)

\(x^2+y^2\)

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

\(=a^2-2b\)

\(b)\)

\(x^3+y^3\)

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

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

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

\(=a^3-3ab\)

\(c)\)

\(x^4+y^4\)

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

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

\(=a^4-4a^2b+2b^2\)

\(d)\)

\(x^5+y^5\)

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

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

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

\(=a^5-3a^3b-2a^3b+6ab^2-ab^2\)

\(=a^5-5a^3b+5ab^2\)

Bài 1:

a: =>x^3-x-6x-6=0

=>x(x-1)(x+1)-6(x+1)=0

=>(x+1)(x-3)(x+2)=0

hay \(x\in\left\{-1;3;-2\right\}\)

b: \(\Leftrightarrow x^2-6x+9+y^2+6y+9=0\)

=>(x-3)^2+(y+3)^2=0

=>x=3 và y=-3

a) 

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

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

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

\(=\left(x+y\right)^5-5xy\left(x^3+2xy^2+2x^2y+y^3\right)=\left(x+y\right)^5-5xy\left(x^3+3xy^2+3x^2y+y^3-xy^2-x^2y\right)\)

\(=\left(x+y\right)^5-5xy\left(\left(x+y\right)^3-xy\left(x+y\right)\right)=a^5-5b\left(a^3-ab\right)\)

giup minh cau b o tren nha

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 : \(\left(x+y\right)^3=1^3=1\)

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

\(\Leftrightarrow x^3+y^3+3xy=1\) ( do x + y = 1 )

Áp dụng BĐT Cauchy-Schwarz ta có:

\(\left(\dfrac{x^3}{y^2}+\dfrac{y^3}{z^2}+\dfrac{z^3}{x^2}\right)\left(x+y+z\right)\ge\left(\dfrac{x^2}{y}+\dfrac{y^2}{z}+\dfrac{z^2}{x}\right)^2\)

Cần chứng minh \(\dfrac{x^2}{y}+\dfrac{y^2}{z}+\dfrac{z^2}{x}\ge x+y+z\)

Dễ thấy;\(VT=\dfrac{x^2}{y}+\dfrac{y^2}{z}+\dfrac{z^2}{x}\ge\dfrac{\left(x+y+z\right)^2}{x+y+z}=x+y+z\)

BĐT được chứng minh

\("="\Leftrightarrow x=y=z\)

\(B=8x^2+2x-8x^3-8x^2+8x^3-2x+3=3\)

\(C=x^3-3x^2+3x-1+x^3+3x^2+3x+1+2x^3-8x=4x^3-2x\)

\(D=\left(x+y-5\right)^2-2\left(x+y-5\right)\left(x+3\right)+\left(x+3\right)^2=\left(x+y-5-x-3\right)^2=\left(y-8\right)^2\)

câu 2. ta có 

a.\(\left(x-y\right)^2=\left(x+y\right)^2-4xy=7^2-4\times12=1\)

b.\(3\left(x^2+y^2\right)-2\left(x^3+y^3\right)=3\left(x+y\right)^2-6xy-2\left(x+y\right)^3+6xy\left(x+y\right)=3-6xy-2+6xy=1\)

Bài 1.

A = x² + 2xy + y² = ( x + y )² = ( -1 )² = 1

B = x² + y² = ( x² + 2xy + y² ) - 2xy = ( x + y )² - 2xy = (-1)² - 2.(-12) = 1 + 24 = 25

C = x³ + 3xy( x + y ) + y³ = ( x³ + y³ ) + 3xy( x + y ) = ( x + y )( x² - xy + y² ) + 3xy( x + y )

                                                                                  = -1( 25 + 12 ) + 3.(-12).(-1)

                                                                                  = -37 + 36

                                                                                  = -1

D = x³ + y³ = ( x³ + 3x²y + 3xy² + y³ ) - 3x²y - 3xy² = ( x + y )³ - 3xy( x + y ) = (-1)³ - 3.(-12).(-1) = -1 - 36 = -37

Bài 2.

M = 3( x² + y² ) - 2( x³ + y³ )

= 3( x² + y² ) - 2( x + y )( x² - xy + y² )

= 3( x² + y² ) - 2( x² - xy + y² )

= 3x² + 3y² - 2x² + 2xy - 2y²

= x² + 2xy + y²

= ( x + y )² = 1² = 1

Bài 3.

x + y = -3

<=> ( x + y )² = 9

<=> x² + 2xy + y² = 9

<=> 29 + 2xy = 9

<=> 2xy = -20

<=> xy = -10

x³ + y³ = ( x³ + 3x²y + 3xy² + y³ ) - 3x²y - 3xy² = ( x + y )³ - 3xy( x + y ) = ( -3 )³ - 3.(-10).(-3) = -27 - 90 = -117

Bài 4.

x - y = 5

<=> ( x - y )² = 25

<=> x² - 2xy + y² = 25

<=> 15 - 2xy = 25

<=> 2xy = -10

<=> xy = -5

x³ - y³ = ( x³ - 3x²y + 3xy² - y³ ) + 3x²y - 3xy² = ( x - y )³ + 3xy( x - y ) = 5³ + 3.(-5).5 = 125 - 75 = 50