Lời giải:
Đặt \(B=\frac{x^2+y^2+z^2}{(y-z)^2+(z-x)^2+(x-y)^2}\)
\(\Leftrightarrow B=\frac{x^2+y^2+z^2}{2(x^2+y^2+z^2-xy-yz-xz)}\)
Lại có
\(x^2+y^2+z^2=(x+y+z)^2-2(xy+yz+xz)=-2(xy+yz+xz)\)
\(\Rightarrow B=\frac{-2(xy+yz+xz)}{2[-2(xy+yz+xz)-(xy+yz+xz)]}=\frac{-2(xy+yz+xz)}{-6(xy+yz+xz)}=\frac{1}{3}\)
1. Cho biết x , y , z # 0 và \(\dfrac{\left(ax+by+cz\right)^2}{x^2+y^2+z^2}=a^2+b^2+c^2\) .
Chứng minh rằng : \(\dfrac{a}{x}=\dfrac{b}{y}=\dfrac{c}{z}\)
2. Rút gọn : \(\dfrac{x^2+y^2+z^2}{\left(y-z\right)^2+\left(z-x\right)^2+\left(x-y\right)^2}\) , biết rằng : x + y + z = 0
3. Cho 3x - y = 3z và 2x + y = 7z . Tính giá trị cua biểu thức :
M = \(\dfrac{x^2-2xy}{x^2+y^2}\) ( x # 0 ; y # 0 )
Cho x + y + z + 0 và x, y, z \(\ne\) 0. Rút gọn :
a/ \(P=\dfrac{x^2+y^2+z^2}{\left(x-y\right)^2+\left(y-z\right)^2+\left(z-x\right)^2}\)
b/ \(Q=\dfrac{\left(x^2+y^2-z^2\right)\cdot\left(y^2+z^2-x^2\right)\cdot\left(z^2+x^2-y^2\right)}{16\cdot x\cdot y\cdot z}\)
Rút gọn phân thức:
1, \(\dfrac{x^2+y^2-1+2xy}{x^2-y^2+1+2x}\)
2, \(\dfrac{x^4-y^4}{x^3+y^3}\)
3, \(\dfrac{x^3+y^3+z^3-3xyz}{\left(x-y\right)^2+\left(x-z\right)^2+\left(y-z\right)^2}\)
4, \(\dfrac{\left(x^2-y^2\right)^3+\left(y^2-z^2\right)^3+\left(z^2-x^2\right)^3}{\left(x-y\right)^3+\left(y-z\right)^3+\left(z-x\right)^3}\)
5, \(\dfrac{x^3-7x+6}{x^2\left(x-3\right)^2+4x\left(3-x\right)^2+4\left(x-3\right)^2}\)
rút gọn biểu thức A= \(\dfrac{x^3-y^3-z^3-3xyz}{\left(x+y\right)^2+\left(y-z\right)^2+\left(x+z\right)^2}\)
Rút gọn phân thức
B= \(\dfrac{x^2\left(y-z\right)+y^2\left(z-x\right)+z^2\left(x-y\right)}{x^2y-x^2z+y^2z-y^3}\)
Rút gọn phân thức
1. \(\dfrac{a^2\left(b-c\right)+b^2\left(c-a\right)+c^2\left(a-b\right)}{ab^2-ac^2-b^3+bc^2}\)
2.\(\dfrac{2x^3-7x^2-12x+45}{3x^3-19x^2+33x-9}\)
3.\(\dfrac{x^3-y^3+z^3+3xyz}{\left(x+y\right)^2+\left(y+z\right)^2+\left(z-x\right)^2}\)
4. \(\dfrac{x^3+y^3+z^3-3xyz}{\left(x-y\right)^2+\left(y-z\right)^2+\left(z-x\right)^2}\)
1.
Cho biết ax+by+cz=0
Rút gọn A=\(\dfrac{bc.\left(y-z\right)^2+ca.\left(z-x\right)^2+ab.\left(x-y\right)^2}{ax^2+by^2+cz^2}\)
\(\dfrac{x^2+y^2+z^2}{\left(y-z\right)^2+\left(z-x\right)^2+\left(x-y\right)^2}\)
Rút gọn phân thức
a,\(\dfrac{\left(x^2-y\right).\left(y+1\right)+x^2y^2-1}{\left(x^2+y\right).\left(y+1\right)+x^2y^2+1}\)
b,\(\dfrac{x^2\left(y-z\right)+y^2\left(z-x\right)+z^2\left(x+y\right)}{x^2y-x^2z+y^2z-y^3}\)
c, \(\dfrac{x^3+3x^2-4}{x^3-3x+2}\)
d , \(\dfrac{x^4+6x^3+9x^2-1}{x^4+6x^3+7x^2-6x+1}\)
