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a: \(A=x^3+3x^2+3x+1-1\)
\(=\left(x+1\right)^3-1\)
\(=100^3-1=999999\)
b: \(B=3\left[\left(x+y\right)^2-2xy\right]-2\left[\left(x+y\right)^3-3xy\left(x+y\right)\right]\)
\(=3\left(1-2xy\right)-2\left(1-3xy\right)\)
\(=3-6xy-2+6xy=1\)
c: \(C=\left(x^3+3x^2y+3xy^2+y^3\right)-3\left(x^2+2xy+y^2\right)+3\left(x+y\right)+2017\)
\(=101^3-3\cdot101^2+3\cdot101+2017\)
\(=101^3-3\cdot101^2+3\cdot101-1+2018\)
\(=100^3+2018=1002018\)
Lời giải:
a)
$(a-b)^3=(a-b)^2.(a-b)=(b-a)^2.-(b-a)=-(b-a)^3$
b)
$(-a-b)^2=[-(a+b)]^2=(-1)^2(a+b)^2=(a+b)^2$
c)
$(x+y)^3=x^3+3x^2y+3xy^2+y^3$
$=x^3-6x^2y+9x^2y-6xy^2+9xy^2+y^3$
$=(x^3-6x^2y+9xy^2)+(y^3-6xy^2+9x^2y)$
$=x(x^2-6xy+9y^2)+y(y^2-6xy+9x^2)$
$=x(x-3y)^2+y(y-3x)^2$
d)
$(x+y)^3-(x-y)^3=x^3+3xy(x+y)+y^3-[x^3-3xy(x-y)-y^3]$
$=2y^3+3xy[(x+y)+(x-y)]=2y^3+6x^2y=2y(y^2+3x^2)$
Lời giải:
a)
$(a-b)^3=(a-b)^2.(a-b)=(b-a)^2.-(b-a)=-(b-a)^3$
b)
$(-a-b)^2=[-(a+b)]^2=(-1)^2(a+b)^2=(a+b)^2$
c)
$(x+y)^3=x^3+3x^2y+3xy^2+y^3$
$=x^3-6x^2y+9x^2y-6xy^2+9xy^2+y^3$
$=(x^3-6x^2y+9xy^2)+(y^3-6xy^2+9x^2y)$
$=x(x^2-6xy+9y^2)+y(y^2-6xy+9x^2)$
$=x(x-3y)^2+y(y-3x)^2$
d)
$(x+y)^3-(x-y)^3=x^3+3xy(x+y)+y^3-[x^3-3xy(x-y)-y^3]$
$=2y^3+3xy[(x+y)+(x-y)]=2y^3+6x^2y=2y(y^2+3x^2)$
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 , Ta có: \(a+b=10\Rightarrow a=10-b\)
\(a+b=10\Rightarrow\left(a+b\right)^2=100\)
\(\Leftrightarrow a^2+2ab+b^2=100\)
\(\Leftrightarrow2ab=100-\left(a^2+b^2\right)=100-52=48\Rightarrow ab=24\)\(\Leftrightarrow\left(10-b\right)b=24\Leftrightarrow10b+b^2-24=0\)
\(\Leftrightarrow b^2+10b+25-49=0\)
\(\Leftrightarrow\left(b+5\right)^2=49\Rightarrow\left[{}\begin{matrix}b+5=7\\b+5=-7\end{matrix}\right.\Rightarrow\left[{}\begin{matrix}b=2\\b=-12\end{matrix}\right.\Rightarrow\left[{}\begin{matrix}a=8\\a=-2\end{matrix}\right.\)b, ta có:
\(\left(x+y\right)=1\Rightarrow\left(x+y\right)^3=1\)
\(\Leftrightarrow x^3+3x^2y+3xy^2+y^3=1\)
\(\Leftrightarrow x^3+3xy\left(x+y\right)+y^3=1\)
\(\Leftrightarrow x^3+3xy+y^3=1\)
c, \(a+b=13\Rightarrow\left(a+b\right)^2=169\)
\(\Leftrightarrow a^2+2ab+b^2=169\Rightarrow a^2+b^2=169-2ab=169-2.9=151\)\(\Rightarrow a^3+b^3=\left(a+b\right)\left(a^2+b^2+ab\right)=13.\left(151+9\right)=2080\)
\(d,x+y=7\Rightarrow\left(x+y\right)^2=49\)
\(\Leftrightarrow x^2+2xy+y^2=49\Rightarrow2xy=49-\left(x^2+y^2\right)=16\Rightarrow xy=8\)
\(\Rightarrow x^3+y^3=\left(x+y\right)\left(x^2-xy+y^2\right)=7.\left(33-8\right)=175\)
\(A=x^3+y^3+3xy=\left(x+y\right)^3-3xy\left(x+y\right)+3xy=1+0=1\)
\(B=\left(x-y\right)^3+3xy\left(x-y\right)-3xy=1\)
\(c,M=a^2-ab+b^2+3ab\left(a^2+b^2\right)+6a^2b^2=3ab\left(a^2+2ab+b^2\right)+a^2-ab+b^2\)
\(=3ab+a^2-ab+b^2=\left(a+b\right)^2=1\)
\(x+y=2;x^2+y^2=10\text{ do đó:}xy=-3\text{ nên }\left(x-y\right)^2=16\text{ do đó: }x-y=4\text{ hoặc }x-y=-4\)
\(\text{giải ra được:}x=3;y=-1\text{ hoặc ngược lại nên }x^3+y^3=-26\text{ hoặc }26\)
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 = x3 + 3x2 + 3x - 899
= (x3 + 3x2 + 3x + 1) - 900
= (x + 1)3 - 900
= (29 + 1)3 - 900 = 303 - 900 = 26100
B = x3 - 6x2 + 12x + 10
= (x3 - 6x2 + 12x - 8) + 18
= (x - 2)3 + 18
= (12 - 2)3 + 18 = 103 + 18 = 1000 + 18 = 1018
c) C = 8x3 - 27y3
= (2x)3 - (3y)3
= (2x - 3y)(4x2 + 6xy + 9y2)
= (2x - 3y)(4x2 - 12xy + 9y2) + (2x - 3y).18xy
= (2x - 3y)(2x - 3y)2 + (2x - 3y).18xy
= (2x - 3y)3 + (2x - 3y).18xy
= 53 + 5.18.4
= 125 - 360
= -235
D = x3 + y3 + 3xy(x2 + y2) + 6x2y2(x + y)
= (x + y)(x2 - xy + y2) + 3x3y + 3xy3 + 6x2y2
= x2 + y2 - xy + 3x3y + 3xy3 + 6x2y2
= (x + y)2 - 3xy + 3x3y + 3xy3 + 6x2y2
= 1 - 3xy(2xy - 1) + 3xy(x2 + y2)
= 1 - 3xy(x2 + y2 + 2xy - 1)
= 1 - 3xy[(x + y)2 - 1]
= 1 - 0 = 1