rút gọn biểu thức
A= (x^2-y)(y+1)+x^2y^2-1
(x^2+y)(Y+1)+x^2y^2+1
B =(y-z)+y^2(z-x)+z^2
x^2y-x^2z+y^2z-y^3
K
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S
16 tháng 8 2016
\(B=\frac{x^2\left(y-z\right)+y^2\left(z-x\right)+z^2\left(x-y\right)}{x^2y-x^2z+y^2z-y^3}\)
\(=\frac{x^2y-x^2z+zy^2-xy^2+z^2x-z^2y}{x^2\left(y-z\right)-y^2\left(y-z\right)}\)
\(=\frac{\left(x^2y-z^2y\right)-\left(xy^2-zy^2\right)-\left(x^2z-z^2x\right)}{\left(x^2-y^2\right)\left(y-z\right)}\)
\(=\frac{\left[y\left(x+z\right)-y^2-xz\right]\left(x-z\right)}{\left(x-y\right)\left(x+y\right)\left(y-z\right)}\)
\(=\frac{\left(xy+zy-y^2-xz\right)\left(x-z\right)}{\left(x-y\right)\left(x+y\right)\left(y-z\right)}\)
\(=\frac{\left[\left(xy-y^2\right)-\left(xz-zy\right)\right]\left(x-z\right)}{\left(x-y\right)\left(x+y\right)\left(y-z\right)}\)
\(=\frac{\left[y\left(x-y\right)-z\left(x-y\right)\right]\left(x-z\right)}{\left(x-y\right)\left(x+y\right)\left(y-z\right)}\)
\(=\frac{\left(y-z\right)\left(x-y\right)\left(x-z\right)}{\left(x-y\right)\left(x+y\right)\left(y-z\right)}\)
\(=\frac{x-z}{x+y}\)
S
16 tháng 8 2016
\(A=\frac{\left(x^2-y\right)\left(y+1\right)+x^2y^2-1}{\left(x^2+y\right)\left(y+1\right)+x^2y^2+1}\)
\(=\frac{x^2y-y^2+x^2-y+x^2y^2-1}{x^2y+y^2+x^2+y+x^2y^2+1}\)
\(=\frac{\left(x^2y+x^2\right)+\left(x^2y^2-y^2\right)-\left(y+1\right)}{\left(x^2y+x^2\right)+\left(x^2y^2+y^2\right)+\left(y+1\right)}\)
\(=\frac{x^2\left(y+1\right)+y^2\left(x^2-1\right)-\left(y+1\right)}{x^2\left(y+1\right)+y^2\left(x^2+1\right)+\left(y+1\right)}\)
\(=\frac{\left(x^2-1\right)\left(y+1\right)+y^2\left(x^2-1\right)}{\left(x^2+1\right)\left(y+1\right)+y^2\left(x^2+1\right)}\)
\(=\frac{\left(x^2-1\right)\left(y^2+y+1\right)}{\left(x^2+1\right)\left(y^2+y+1\right)}\)
\(=\frac{x^2-1}{x^2+1}\)
E
2 tháng 9 2021
(x - y + z)^2 + (z - y)^2 + (x - y + z)(2y -2z)
\(<=>(x-y+z)^2+2(x-y+z)(y-z)+(z-y)^2\)
\(<=> (x-y+z+z-y)^2<=> ( x-2y-2z)^2\)
2 tháng 9 2021
\(\left(x+y+z\right)^2+\left(z-y\right)^2+\left(x-y+z\right)\left(2y-2z\right)\)
\(=\left(x+y+z+y-z\right)^2\)
\(=\left(x+2y\right)^2\)
21 tháng 9 2023
`#3107`
`a)`
`A=`\(3x^4 + \dfrac{1}3xyz - 3x^4 - \dfrac{4}3xyz + 2x^2y - 6z\)
`= (3x^4 - 3x^4) + (1/3xyz - 4/3xyz) + 2x^2y - 6z`
`= -xyz + 2x^2y - 6z`
Thay `x = 1; y = 3` và `z = 1/3` vào A
`A = -1*3*1/3 + 2*1^2*3 - 6*1/3`
`= -1 + 6 - 2`
`= 6 - 3`
`= 3`
Vậy, `A=3`
`b)`
`B=`\(4x^3 - \dfrac{2}7xyz - 4x^3 - \dfrac{4}3xyz + 4x^2y\)
`= (4x^3 - 4x^3) + (-2/7xyz - 4/3xyz) + 4x^2y`
`= -34/21 xyz + 4x^2y`
Thay `x = -1; y = 2` và `z = -1/2` vào B
`B = -34/21*(-1)*2*(-1/2) + 4*(-1)^2 * 2`
`= -34/21 + 8`
`= 134/21`
Vậy, `B = 134/21`
`c)`
`C=`\(4x^2 + \dfrac{1}2xyz - \dfrac{2}3xy^2z - 5x^2yz + \dfrac{3}4xyz\)
`= 4x^2 + (1/2xyz + 3/4xyz) - 2/3xy^2z - 5x^2yz `
`= 4x^2 + 5/4xyz - 2/3xy^2z - 5x^2yz`
Ta có:
`|y| = 2`
`=> y = +-2`
Thay `x = -1; y = 2` và `z = 1/2` vào C
`4*(-1)^2 + 5/4*(-1)*2*1/2 - 2/3*(-1)*2^2*1/2 - 5*(-1)^2*2*1/2`
`= 4 - 5/4 + 4/3 - 5`
`= -11/12`
Vậy, với `x = -1; y = 2; z = 1/2` thì `B = -11/12`
Thay `x = -1; y = -2; z = 1/2`
`B = 4*(-1)^2 + 5/4*(-1)*(-2)*1/2 - 2/3*(-1)*(-2)^2*1/2 - 5*(-1)^2*(-2)*1/2`
`= 4 + 5/4 + 4/3 + 5`
`= 139/12`
Vậy, với `x = -1; y = -2; z = 1/2` thì `B = 139/12.`
S
24 tháng 11 2018
a, Xét tử thức \(x^2\left(y-z\right)+y^2\left(z-x\right)+z^2\left(x-y\right)\)
\(=x^2\left(y-z\right)-y^2\left(x-z\right)+z^2\left[\left(x-z\right)-\left(y-z\right)\right]\)
\(=x^2\left(y-z\right)-y^2\left(x-z\right)+z^2\left(x-z\right)-z^2\left(y-z\right)\)
\(=\left(x^2-z^2\right)\left(y-z\right)-\left(y^2-z^2\right)\left(x-z\right)\)
\(=\left(x-z\right)\left(x+z\right)\left(y-z\right)-\left(y-z\right)\left(y+z\right)\left(x-z\right)\)
\(=\left(x-z\right)\left(xy-xz+yz-z^2-y^2-yz+yz+z^2\right)\)
\(=\left(x-z\right)\left(xy-xz+yz-y^2\right)=\left(x-z\right)\left[x\left(y-z\right)-y\left(y-z\right)\right]\)
\(=\left(x-z\right)\left(x-y\right)\left(y-z\right)\)
Mẫu thức \(x^2y-x^2z+y^2z-y^3=x^2\left(y-z\right)-y^2\left(y-z\right)=\left(x-y\right)\left(x+y\right)\left(y-z\right)\)
Vậy \(\frac{x^2\left(y-z\right)+y^2\left(z-x\right)+z^2\left(x-y\right)}{x^2y-x^2z+y^2z-y^3}=\frac{\left(x-y\right)\left(y-z\right)\left(x-z\right)}{\left(x-y\right)\left(x+y\right)\left(y-z\right)}=\frac{x-z}{x+y}\)
b, \(\frac{x^5+x+1}{x^3+x^2+x}=\frac{x^5-x^2+x^2+x+1}{x\left(x^2+x+1\right)}=\frac{x^2\left(x-1\right)\left(x^2+x+1\right)+x^2+x+1}{x\left(x^2+x+1\right)}=\frac{\left(x^2+x+1\right)\left(x^3-x^2+1\right)}{x\left(x^2+x+1\right)}=\frac{x^3-x^2+1}{x}\)
BN
17 tháng 6 2018
a,A=5x2z-10xyz+5y2z
=5z(x2-2xy+y2)
=5z(x-y)2
Thay x=124,y=24,z=2 vào A ta được:
A=5.2(124-24)2=10.1002=10000
b,B=2x2+2y2-x2z+z-y2z-2
=2(x2+y2)-z(x2+y2)+(z-2)
=(2-z)(x2+y2)-(2-z)
=(2-z)(x2+y2-1)
Thay x=1,y=1,z=-1 vào B
B=(2+1)(12+12-1)=3
c, C=x2-y2+2y-1
=x2-(y2-2y+1)
=x2-(y-1)2
=(x-y+1)(x+y-1)
=(75-26+1)(75+26-1)
=50.100=5000
DD
Đoàn Đức Hà
Giáo viên
4 tháng 8 2021
\(\frac{2}{x+y+z}=\frac{x}{2y+2z+1}=\frac{y}{2x+2z+1}=\frac{z}{2x+2y-2}=\frac{x+y+z}{4\left(x+y+z\right)}=\frac{1}{4}\)
\(\Rightarrow\hept{\begin{cases}2y+2z+1=4x\\2x+2z+1=4y\\x+y+z=8\end{cases}}\Leftrightarrow\hept{\begin{cases}x=y=\frac{17}{6}\\z=\frac{7}{3}\end{cases}}\)
12 tháng 11 2017
ap dung bdt \(x^{m+n}+y^{m+n}\ge x^my^n+x^ny^m\) (bn tu cm )
\(\Rightarrow x^7+y^7=x^{3+4}+y^{3+4}\ge x^3y^4+x^4y^3\)
\(\Rightarrow\frac{x^2y^2}{x^2y^2+x^7+y^7}\le\frac{x^2y^2}{x^2y^2\left(1+xy^2+x^2y\right)}=\frac{1}{1+x^2y+y^2x}=\frac{1}{xyz+x^2y+y^2x}=\frac{1}{xy\left(x+y+z\right)}=\)
=\(\frac{z}{xyz\left(x+y+z\right)}=\frac{z}{x+y+z}\)
ttu \(P\le\frac{x+y+z}{x+y+z}=1\) đầu = xảy ra khi x=y=z=1
