Cho x+y+z=0
Chứng minh 2(x^5+y^5+z^5) = 5xyz(x^2+y^2+z^2)
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KN
0
HP
1
27 tháng 2 2018
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)
L
28 tháng 8 2021
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)
NM
1
AH
Akai Haruma
Giáo viên
27 tháng 8 2023
Lời giải:
Ta có:
$x^3+y^3+z^3=(x+y)^3-3xy(x+y)+z^3=(-z)^3-3xy(-z)+z^3$
$=(-z)^3+3xyz+z^3=3xyz$
Khi đó:
$2(x^5+y^5+z^5)=2[(x^3+y^3+z^3)(x^2+y^2+z^2)-(x^3y^2+x^3z^2+y^3x^2+y^3z^2+z^3x^2+z^3y^2)]$
$=2[3xyz(x^2+y^2+z^2)-x^2y^2(x+y)-y^2z^2(y+z)-z^2x^2(z+x)]$
$=6xyz(x^2+y^2+z^2)-2[x^2y^2(-z)+y^2z^2(-x)+z^2x^2(-y)]$
$=6xyz(x^2+y^2+z^2)+2(x^2y^2z+y^2z^2x+x^2x^2y)$
$=6xyz(x^2+y^2+z^2)+2xyz(xy+yz+xz)$
$=6xyz(x^2+y^2+z^2)+xyz[(x+y+z)^2-(x^2+y^2+z^2)]$
$=6xyz(x^2+y^2+z^2)+xyz[0-(x^2+y^2+z^2)]$
$=6xyz(x^2+y^2+z^2)-xyz(x^2+y^2+z^2)=5xyz(x^2+y^2+z^2)$
Ta có đpcm.
HT
0
TT
1
18 tháng 8 2017
\(y+z=-x\)
\(\Leftrightarrow\left(y+z\right)^5=-x^5\)
\(\Leftrightarrow y^5+5y^4z+10y^3z^2+10y^2z^3+5yz^4+z^5+x^5=0\)
\(\Leftrightarrow x^5+y^5+z^5+5yz\left(y^3+2y^2z+2yz^2+z^3\right)=0\)
\(\Leftrightarrow x^5+y^5+z^5+5yz\left[\left(y+z\right)\left(y^2-yz-z^2\right)+2yz\left(y+z\right)\right]=0\)
\(\Leftrightarrow x^5+y^5+z^5+5yz\left(y+z\right)\left(y^2+yz+z^2\right)=0\)
\(\Leftrightarrow2\left(x^5+y^5+z^5\right)-5xyz\left[\left(y^2+2yz+z^2\right)+y^2+z^2\right]=0\)
\(\Leftrightarrow2\left(x^5+y^5+z^5\right)=5xyz\left(x^2+y^2+z^2\right)\)(đpcm)
SL
0
29 tháng 3 2020
Ta có: 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)
NQ
2 tháng 4 2020
Từ x+y+z=0 => y+z=-x => (y+z)5=-x5
=> \(y^5+5y^4z+10y^2z^2+10y^2z^3+5yz^4+z^5=-x^5\)
\(\Rightarrow\left(x^5+y^5+z^5\right)+5yz\left(y^3+2y^2z+2yz^2+z^3\right)=0\)
\(\Rightarrow\left(x^5+y^5+z^5\right)+5yz\left[\left(y+z\right)\left(y^2-yz+x^2\right)\right]=0\)
\(\Rightarrow\left(x^5+y^5+z^5\right)+5yz\left(y+z\right)\left(y^2+yz+z^2\right)=0\)
\(\Rightarrow2\left(x^5+y^5+z^5\right)-5xyz\left[\left(y^2+2yz+z^2\right)+y^2+z^2\right]=0\)
\(\Rightarrow2\left(x^5+y^5+z^5\right)=5xyz\left[\left(y+z\right)^2+y^2+z^2\right]\) (đpcm)