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a) \(A=x^2-2xy+y^2=\left(x-y\right)^2=\left(-3\right)^2=9\)
b) \(B=x^2+y^2=x^2-y^2+2xy-2xy=\left(x-y\right)^2+2xy=9+2.10=29\)
c) \(C=x^3-3x^2y+3xy^2-y^3=\left(x-y\right)^3=\left(-3\right)^3=-27\)
d) \(D=x^3-y^3=\left(x-y\right)^3+3xy\left(x-y\right)=-27+3.10.\left(-3\right)=-27-90=-117\)
a) \(A=x\left(x+2\right)+y\left(y-2\right)-2xy+37\)
\(A=x^2+2x+y^2-2y-2xy+37\)
\(A=\left(x^2-2xy+y^2\right)+\left(2x-2y\right)+37\)
\(A=\left(x-y\right)^2+2\left(x-y\right)+37\)
\(A=\left(x-y\right)^2+2\left(x-y\right)+1+36\)
\(A=\left(x-y+1\right)^2+36\)
Thay x - y = 7 vào A
\(A=\left(7+1\right)^2+36\)
\(A=8^2+36\)
\(A=64+36\)
\(A=100\)
b) \(B=x^3+x^2-y^3+y^2+xy-3x^2y+3xy^2-3xy-9\)
\(B=\left(x^3-3x^2y+3xy^2-y^3\right)+\left(x^2+xy-3xy+y^2\right)-9\)
\(B=\left(x-y\right)^3+\left(x^2-2xy+y^2\right)-9\)
\(B=\left(x-y\right)^3+\left(x-y\right)^2-9\)
Thay x - y = 7 vào B
\(B=7^3+7^2-9\)
\(B=343+49-9\)
\(B=383\)
c) \(C=x^3-x^2-y^3-y^2-3xy\left(x-y\right)+2xy\)
\(C=\left[x^3-y^3-3xy\left(x-y\right)\right]-\left(x^2-2xy+y^2\right)\)
\(C=\left(x-y\right)^3-\left(x-y\right)^2\)
Thay x - y = 7 vào C
\(C=7^3-7^2\)
\(C=343-49\)
\(C=294\)
d) \(D=x^2\left(x+1\right)-y^2\left(y-1\right)+xy-3xy\left(x-y+1\right)-95\)
\(D=x^3+x^2-y^3+y^2+xy-3x^2y+3xy^2-3xy-95\)
\(D=\left(x^3-3x^2y+3xy^2-y^3\right)+\left(x^2-2xy+y^2\right)-95\)
\(D=\left(x-y\right)^3+\left(x-y\right)^2-95\)
Thay x - y = 7 vào D
\(D=7^3+7^2-95\)
\(D=343+49-95\)
\(D=297\)
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
\(A=x^2-2xy+y^2=\left(x-y\right)^2=\left(-3\right)^2=9\)
\(B=x^2+y^2=\left(x-y\right)^2+2xy=9+10.2=29\)
\(C=x^3-3x^2y+3xy^2-y^3=\left(x-y\right)^3=\left(-3\right)^3=-27\)
\(D=x^3-y^3=\left(x-y\right)\left(x^2+xy+y^2\right)=\left(-3\right)\left[x^2-2xy+y^2+3xy\right]=\left(-3\right)\left(\left(-3\right)^2.3.10\right)=-3.270=-810\)
a ) có \(x^2+y^2+4x-2xy+4y+2019=\left(x-y\right)^2+4\left(x-y\right)+2019=49+28+2019=2096\)
b) \(x^3-3xy\left(x-y\right)-y^3-x^2+2xy-y^2=\left(x-y\right)^3-\left(x-y\right)^2=343-49=294\)
c)\(x^2\left(x+1\right)-y^2\left(y-1\right)+xy-3xy\left(x-y+1\right)=x^3-y^3+x^2+y^2+xy-3x^2y+3xy^2-3xy=\left(x-y\right)^3+\left(x-y\right)^2=343+49=392\)
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
a) \(A=x^2+2xy+y^2=\left(x+y\right)^2=\left(-1\right)^2=1\)
b) \(B=x^2+y^2=x^2+y^2+2xy-2xy=\left(x+y\right)^2-2.\left(-12\right)=1-\left(-24\right)=25\)
c) \(C=x^3+3x^2y+3xy^2+y^3=\left(x+y\right)^3=\left(-1\right)^3=-1\)
d, D = x3 + y3 = ( x + y)3 - 3xy( x + y) = -1 - 36 = -37