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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, \(\frac{x^5+x+1}{x^3+x^2+x}=\frac{x^5-x^2+x^2+x+1}{x\left(x^2+x+1\right)}=\frac{x^2\left(x-1\right)\left(x^2+x+1\right)+x^2+x+1}{x\left(x^2+x+1\right)}=\frac{\left(x^2+x+1\right)\left(x^3-x^2+1\right)}{x\left(x^2+x+1\right)}=\frac{x^3-x^2+1}{x}\)
Đây nha c ơi
a: x-y-z=0
=>x=y+z; y=x-z; z=x-y
\(K=\dfrac{x-z}{x}\cdot\dfrac{y-x}{y}\cdot\dfrac{z+y}{z}=\dfrac{y\cdot\left(-z\right)\cdot x}{xyz}=-1\)
b: Tham khảo:
Xét các biểu thức :
\(x^3+y^3+z^3=x^3+y^3+\left(-x-y\right)^3=\left(x+y\right)\left(x^2-xy+y^2\right)-\left(x+y\right)\left(x^2+2xy+y^2\right)\)
\(=\left(x+y\right)\left(-3xy\right)=-3xy.\left(-z\right)=3xyz\)
\(x^2+y^2+z^2=x^2+y^2+\left(-x-y\right)^2=2\left(x^2+y^2+xy\right)\)
Do đó VT có giá trị là \(5.\left(3xyz\right).2\left(x^2+y^2+xy\right)=30xyz\left(x^2+y^2+xy\right)\)
Xét VP:
\(x^5+y^5+z^5=\left(x^5+y^5\right)+\left(-x-y\right)^5\)
\(=x^5+y^5-\left(x^5+5x^4y+10x^3y^2+10x^2y^3+5xy^4+y^5\right)\)
\(=-5xy\left(x^3+2x^2y+2xy^2+y^3\right)\)
\(=-5xy.\left[\left(x+y\right)^3-xy\left(x+y\right)\right]\)
\(=-5xy\left(x+y\right)\left(x^2+2xy+y^2-xy\right)\)
\(=5xyz\left(x^2+xy+y^2\right)\)
Do đó VP là \(30xyz\left(x^2+y^2+xy\right)\)
Suy ra điều phải chứng minh.