Cho x+y+z=0. CMR 2(x^5+y^5+z^5)=5xyz(x^2+y^2+z^2)
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LV
2 tháng 2 2017
Vì x+y+z=0
=>x+y=-z =>(x+y)^5=-z^5
hay x^5+y^5+5(x^4y+xy^4+2x³y²+2x²y³+)=-z^5
<=>x^5+y^5+z^5+5xy(x³+y³+2x²y+2x²y)=0
<=>x5+y^5+z^5+5xy(x+y)(x²-xy+y²+2xy)=0
<=>x^5+y^5+z^5-5xyz(x²+xy+y²)=0
<=>x^5+y^5+z^5=5xyz(x²+xy+y²)
<=>2(x^5+y^5+z^5)=5xyz(2x²+2xy+2y²)
<=>2(x^5+y^5+z^5)=5xyz[x²+y²+(x+y)²]
<=>2(x^5+y^5+z^5)=5xyz(x³+y²+z²)
TC
0
24 tháng 4 2018
\(y+z=-x\)
\(\Leftrightarrow\left(y+z\right)^5=-x^5\)
Áp dụng nhị thức Newton :
\(\Leftrightarrow y^5+5y^4z+10y^3z^2+10y^2z^3+5yz^4+z^5+x^5=0\)
\(\Leftrightarrow x^5+y^5+z^5+5yz\left(y^3+2y^2z+2yz^2+z^3\right)=0\)
\(\Leftrightarrow x^5+y^5+z^5+5yz\left(\left(y+z\right)\left(y^2-yz+z^2\right)+2yz\left(y+z\right)\right)=0\)
\(\Leftrightarrow x^5+y^5+z^5+5yz\left(y+z\right)\left(y^2+yz+z^2\right)=0\)
\(\Leftrightarrow2\left(x^5+y^5+z^5\right)-5xyz\left(\left(y^2+2yz+z^2\right)+y^2+z^2\right)=0\)
\(\Leftrightarrow2\left(x^5+y^5+z^5\right)=5xyz\left(x^2+y^2+z^2\right)\left(dpcm\right)\)
A
0
TH
6
27 tháng 3 2019
Ta có: x + y + z = 0 <=> y + z = -x
(y+z)5 = (-x)5
y5 + z5 + 5y4z + 10y3z2 + 10y2z3 + 5yz4 = -x5
y5 + z5 + 5y4z + 10y3z2 + 10y2z3 + 5yz4 + x5 = 0
x5 + y5 + z5 +5xyz[ y3 + 2y2z + 2yz2 + z3 ] = 0
x5 + y5 + z5 + 5xyz[(y+z)(y2 -yz -z2)+ 2yz(x+z)] = 0
x5 + y5 + z5 +5xyz[(y+z)(y2 +yz + z2)] = 0
2.(x5 + y5 + z5) + 5xyz(y+z)(y2+yz+z2) - (x5 + y5 + z5) = 0
2(x5 + y5 + z5) - 5xyz[(y2+2yz+z2)+y2+z2] = 0
2(x5 + y5 + z5) = 5xyz[(y+z)2 + y2 + z2]
2(x5 + y5 + z5) = 5xyz[(-x)2 + y2 + z2]
2(x5 + y5 + z5) = 5xyz(x2 + y2 + z2).
ND
3
N
30 tháng 7 2015
\(y+z=-x\)
\(\left(y+z\right)^5=-x^5\)
\(y^5+5y^4z+10y^3z^2+10y^2z^3+5yz^4+z^5+x^5=0\)
\(x^5+y^5+z^5+5yz\left(y^3+2y^2z+2yz^2+z^3\right)=0\)
\(x^5+y^5+z^5+5yz\left(\left(y+z\right)\left(y^2-yz+z^2\right)+2yz\left(y+z\right)\right)=0\)
\(x^5+y^5+z^5+5yz\left(y+z\right)\left(y^2+yz+z^2\right)=0\)
\(2\left(x^5+y^5+z^5\right)-5xyz\left(\left(y^2+2yz+z^2\right)+y^2+z^2\right)=0\)
\(2\left(x^5+y^5+z^5\right)=5xyz\left(x^2+y^2+z^2\right)\)
D
8 tháng 9 2018
Ta có: \(y+z=-x\)
\(\left(y+z\right)^5=-x^5\)
\(y^5+5y^4z+10y^3z^2+10y^2z^3+5yz^4+z^5+x^5=0\)
\(x^5+y^5+z^5+5yz\left(y^3+2y^2z+2yz^2+z^3\right)=0\)
\(x^5+y^5+z^5+5yz\left(\left(y+z\right)\left(y^2-yz+z^2\right)+2yz\left(y+z\right)\right)=0\)
\(x^5+y^5+z^5+5yz\left(y+z\right)\left(y^2+yz+z^2\right)=0\)
\(2\left(x^5+y^5+z^5\right)-5xyz\left(\left(y^2+2yz+z^2\right)+y^2+z^2\right)=0\)
\(2\left(x^5+y^5+z^5\right)=5xyz\left(x^2+y^2+z^2\right)\)
x + y + z = 0
⇒x3+y3+z3=3xyz⇒x3+y3+z3=3xyz
⇒(x3+y3+z3)(x2+y2+z2)=3xyz(x2+y2+z2)⇒(x3+y3+z3)(x2+y2+z2)=3xyz(x2+y2+z2)
⇒x5+y5+z5+x2y2(x+y)+y2z2(y+z)+z2x2(z+x)=3xyz(x2+y2+z2)⇒x5+y5+z5+x2y2(x+y)+y2z2(y+z)+z2x2(z+x)=3xyz(x2+y2+z2)
⇒x5+y5+z5−xyz(xy+yx+zx)=3xyz(x2+y2+z2)⇒x5+y5+z5−xyz(xy+yx+zx)=3xyz(x2+y2+z2)
⇒2(x5+y5+z5)=5xyz(x2+y2+z2)