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Từ giả thiết: \(x+y+z=0\Rightarrow x+y=-z\)
\(\Leftrightarrow\left(x+y\right)^3=\left(-c\right)^3\)
\(\Leftrightarrow x^3+3x^2y+3xy^2+y^3=-c^3\)
\(\Leftrightarrow x^3+y^3+z^3=-3xy\left(x+y\right)\)
\(\Leftrightarrow x^3+y^3+z^3=3xyz\) (1)
Nhận cả 2 vế của (1) với \(x^2+y^2+z^2\) ta được:
\(3xyz\left(x^2+y^2+z^2\right)=\left(x^2+y^2+z^2\right)\left(x^3+y^3+z^3\right)=x^5+x^3\left(y^2+z^2\right)+y^5+y^3\left(x^2+z^2\right)+z^5+z^3\left(x^2+y^2\right)\left(2\right)\)Do x + y + z =0 \(\Rightarrow y+z=-x\Rightarrow\left(y+z\right)^2=x^2\Leftrightarrow y^2+z^2=x^2-2yz\)Tương tự ta có:
\(x^2+y^2=z^2-2xy;x^2+z^2=y^2-2xz\)
Thay vào (2) ta được:
\(3xyz\left(x^2+y^2+z^2\right)=x^5+y^5+z^5+x^3\left(x^2-2yz\right)+y^3\left(y^2-2xz\right)+z^3\left(z^2-2xy\right)\)\(=2\left(x^5+y^5+z^5\right)-2xyz\left(x^2+y^2+z^2\right)\)
\(\Leftrightarrow2\left(x^5+y^5+z^5\right)=5xyz\left(x^2+y^2+z^2\right)\left(đpcm\right)\)
Ta có: \(x+y+z=0\Leftrightarrow x+y=-z\)
\(\Leftrightarrow\left(x+y\right)^3=-z^3\Leftrightarrow x^3+3xy\left(x+y\right)+y^3=-z^3\)
\(\Leftrightarrow x^3-3xyz+y^3=-z^3\Leftrightarrow x^3+y^3+z^3=3xyz\)
Do đó \(3xyz\left(x^2+y^2+z^2\right)=\left(x^3+y^3+z^3\right)\left(x^2+y^2+z^2\right)\)
\(=x^5+y^5+z^5+x^3\left(y^2+z^2\right)+y^3\left(z^2+x^2\right)+z^3\left(x^2+y^2\right)\) (*)
Mà \(x^2+y^2=\left(x+y\right)^2-2xy=\left(-z\right)^2-2xy=z^2-2xy\) (vì x + y = -z) (1)
Tương tự, ta có: \(y^2+z^2=x^2-2yz\left(2\right);z^2+x^2=y^2-2zx\left(3\right)\)
Thay (1);(2);(3) vào (*) ta được:
\(3xyz\left(x^2+y^2+z^2\right)=x^5+y^5+z^5+x^3\left(x^2-2yz\right)+y^3\left(y^2-2zx\right)+z^3\left(z^2-2xy\right)\)
\(=x^5+y^5+z^5+x^5-2x^3yz+y^5-2xy^3z+z^5-2xyz^3\)
\(=2\left(x^5+y^5+z^5\right)-2xyz\left(x^2+y^2+z^2\right)\)
\(\Leftrightarrow3xyz\left(x^2+y^2+z^2\right)+2xyz\left(x^2+y^2+z^2\right)=2\left(x^5+y^5+z^2\right)-2xyz\left(x^2+y^2+z^2\right)+2xyz\left(x^2+y^2+z^2\right)\)
\(\Leftrightarrow5xyz\left(x^2+y^2+z^2\right)=2\left(x^5+y^5+z^5\right)\left(đpcm\right)\)
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)
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)
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Vì bài dài nên mình sẽ tách ra nhé.
1a. Ta có:
$x^2+y^2+z^2=(x+y+z)^2-2(xy+yz+xz)=-2(xy+yz+xz)$
$x^3+y^3+z^3=(x+y+z)^3-3(x+y)(y+z)(x+z)=-3(x+y)(y+z)(x+z)$
$=-3(-z)(-x)(-y)=3xyz$
$\Rightarrow \text{VT}=-30xyz(xy+yz+xz)(1)$
------------------------
$x^5+y^5=(x^2+y^2)(x^3+y^3)-x^2y^2(x+y)$
$=[(x+y)^2-2xy][(x+y)^3-3xy(x+y)]-x^2y^2(x+y)$
$=(z^2-2xy)(-z^3+3xyz)+x^2y^2z$
$=-z^5+3xyz^3+2xyz^3-6x^2y^2z+x^2y^2z$
$=-z^5+5xyz^3-5x^2y^2z$
$\Rightarrow 6(x^5+y^5+z^5)=6(5xyz^3-5x^2y^2z)$
$=30xyz(z^2-xy)=30xyz[z(-x-y)-xy]=-30xyz(xy+yz+xz)(2)$
Từ $(1);(2)$ ta có đpcm.
1b.
$x^4+y^4=(x^2+y^2)^2-2x^2y^2=[(x+y)^2-2xy]^2-2x^2y^2$
$=(z^2-2xy)^2-2x^2y^2=z^4+2x^2y^2-4xyz^2$
$x^3+y^3=(x+y)^3-3xy(x+y)=-z^3+3xyz$
Do đó:
$x^7+y^7=(x^4+y^4)(x^3+y^3)-x^3y^3(x+y)$
$=(z^4+2x^2y^2-4xyz^2)(-z^3+3xyz)+x^3y^3z$
$=7x^3y^3z-14x^2y^2z^3+7xyz^5-z^7$
$\Rightarrow \text{VT}=7x^3y^3z-14x^2y^2z^3+7xyz^5$
$=7xyz(x^2y^2-2xyz^2+z^4)$
$=7xyz(xy-z^2)$
$=7xyz[xy+z(x+y)]^2=7xyz(xy+yz+xz)^2$
$=7xyz[x^2y^2+y^2z^2+z^2x^2+2xyz(x+y+z)]$
$=7xyz(x^2y^2+y^2z^2+z^2x^2)$ (đpcm)
Ta có: x+y+z=0 => x3+y3+z3=3xyz (tự c/m)
Mặt khác \(x+y+z=0\Leftrightarrow x+y=-z\Leftrightarrow\left(x+y\right)^2=\left(-z\right)^2\)
\(\Leftrightarrow x^2+2xy+y^2=z^2\Leftrightarrow x^2+y^2=z^2-2xy\)
Tương tự ta cũng có: \(y^2+z^2=x^2-2yz;z^2+x^2=y^2-2zx\)
Lại có: \(\left(x^3+y^3+z^3\right)\left(x^2+y^2+z^2\right)\)
\(=x^5+x^3y^2+x^3z^2+y^3x^2+y^5+y^3z^2+z^3x^2+z^3y^2+z^5\)
\(=x^5+y^5+z^5+x^3\left(y^2+z^2\right)+y^3\left(x^2+z^2\right)+z^3\left(x^2+y^2\right)\)
\(=x^5+y^5+z^5+x^3\left(x^2-2yz\right)+y^3\left(y^2-2xz\right)+z^3\left(z^2-2xy\right)\)
\(=x^5+y^5+z^5+x^5-2x^3yz+y^5-2xy^3z+z^5-2xyz^3\)
\(\Rightarrow3xyz\left(x^2+y^2+z^2\right)=2\left(x^5+y^5+z^5\right)-2xyz\left(x^2+y^2+z^2\right)\)
\(\Rightarrow5xyz\left(x^2+y^2+z^2\right)=2\left(x^5+y^5+z^5\right)\) (đpcm)
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.