\(\text{1) }\dfrac{x^7+x^6+x^5+x^4+x^3+x^2+x+1}{x^2-1}\\ =\dfrac{\left(x^7+x^6\right)+\left(x^5+x^4\right)+\left(x^3+x^2\right)+\left(x+1\right)}{\left(x-1\right)\left(x+1\right)}\\ =\dfrac{x^6\left(x+1\right)+x^4\left(x+1\right)+x^2\left(x+1\right)+\left(x+1\right)}{\left(x-1\right)\left(x+1\right)}\\ =\dfrac{\left(x^6+x^4+x^2+1\right)\left(x+1\right)}{\left(x-1\right)\left(x+1\right)}\\ =\dfrac{x^6+x^4+x^2+1}{x-1}\)

\(\text{3) }\dfrac{x^2+y^2+z^2-2xy+2xz-2yz}{x^2-2xy+y^2-z^2}\\ =\dfrac{\left(x^2-2xy+y^2\right)+\left(2xz-2yz\right)+z^2}{\left(x^2-2xy+y^2\right)-z^2}\\ =\dfrac{\left(x-y\right)^2+2\left(x-y\right)z+z^2}{\left(x-y\right)^2-z^2}\\ =\dfrac{\left(x-y+z\right)^2}{\left(x-y+z\right)\left(x-y-z\right)}\\ =\dfrac{x-y+z}{x-y-z}\)

Câu 1 :

Ta có :

\(\dfrac{x^7+x^6+x^5+x^4+x^3+x^2+x+1}{x^2-1}\)

\(=\dfrac{x^6\left(x+1\right)+x^4\left(x+1\right)+x^2\left(x+1\right)+\left(x+1\right)}{\left(x-1\right)\left(x+1\right)}\)

\(=\dfrac{\left(x+1\right)\left(x^6+x^4+x^2+1\right)}{\left(x-1\right)\left(x+1\right)}\)

\(=\dfrac{\left(x^6+x^4+x^2+1\right)}{\left(x-1\right)}\)

\(=\dfrac{x^4\left(x^2+1\right)+\left(x^2+1\right)}{\left(x-1\right)}\)

\(=\dfrac{\left(x^2+1\right)\left(x^4+1\right)}{x-1}\)

Câu 2 : Ta có VT :

\(\dfrac{2x^2+3xy+y^2}{2x^3+x^2y-2xy^2-y^3}\)

\(=\dfrac{2x^2+2xy+xy+y^2}{x^2\left(2x+y\right)-y^2\left(2x+y\right)}\)

\(=\dfrac{2x\left(x+y\right)+y\left(x+y\right)}{\left(2x+y\right)\left(x^2-y^2\right)}\)

\(=\dfrac{\left(x+y\right)\left(2x+y\right)}{\left(x+y\right)\left(x-y\right)\left(2x+y\right)}\)

\(=\dfrac{1}{x-y}=VP\left(đpcm\right)\)

Câu 3 :

\(\)Ta có :

\(\dfrac{x^2+y^2+z^2-2xy+2xz-2yz}{x^2-2xy+y^2-z^2}\)

\(=\dfrac{\left(x-y+z\right)^2}{\left(x-y\right)^2-z^2}=\dfrac{\left(x-y+z\right)\left(x-y+z\right)}{\left(x-y-z\right)\left(x-y+z\right)}\)

\(=\dfrac{\left(x-y+z\right)}{\left(x-y-z\right)}\)

\(\)

\(\dfrac{x^2+y^2+z^2-2xy+2xz-2yz}{x^2-2xy+y^2-z^2}\)

\(=\dfrac{\left(-x+y-z\right)^2}{\left(x-y\right)^2-z^2}\)

\(=\dfrac{\left[-\left(x-y+z\right)\right]^2}{\left(x-y-z\right)\left(x-y+z\right)}\)

\(=\dfrac{x-y+z}{x-y-z}\)

\(\frac{x^2+y^2+z^2-2xy+2xz-2yz}{x^2-2xy+y^2-z^2}\)

\(=\frac{\left(x-y+z\right)^2}{\left(x-y\right)^2-z^2}\)

\(=\frac{\left(x-y+z\right)^2}{\left(x-y-z\right)\left(x-y+z\right)}\)

\(=\frac{x-y+z}{x-y-z}\)

c) hang dang thuc ( x -y+z)^2

o duoi phan h hang dang thuc luon

a) phan h nhan tu ra sao cho co tử la (x-1)(3x^2 -4x +1)

mau la (x-1)(2x^2 -x-3)

 b ) k nhin dc de

\(\frac{\left(x-y\right)^3+3xy.\left(x+y\right)+y^3}{x-6y}\)

\(=\frac{x^3-3x^2y+3xy^2-y^3+3x^2y+3xy^2+y^3}{x-6y}\)

\(=\frac{x^3+\left(-3x^2y+3x^2y\right)+\left(3xy^2+3xy^2\right)+\left(-y^3+y^3\right)}{x-6y}\)

\(=\frac{x^3+6xy^2}{x-6y}\)

\(\frac{3x^3-7x^2+5x-1}{2x^3-x^2-4x+3}\)

\(=\frac{3x^3-3x^2-4x^2+4x+x-1}{2x^3-2x^2+x^2-x-3x+3}\)

\(=\frac{3x^2.\left(x-1\right)-4x.\left(x-1\right)+\left(x-1\right)}{2x^2.\left(x-1\right)+x.\left(x-1\right)-3.\left(x-1\right)}\)

\(=\frac{\left(x-1\right).\left(3x^2-4x+1\right)}{\left(x-1\right).\left(2x^2+x-3\right)}\)

\(=\frac{3x^2-3x-x+1}{2x^2-2x+3x-3}\)

\(=\frac{3x.\left(x-1\right)-\left(x-1\right)}{2x.\left(x-1\right)+3.\left(x-1\right)}\)

\(=\frac{\left(x-1\right).\left(3x-1\right)}{\left(x-1\right).\left(2x+3\right)}\)

\(=\frac{3x-1}{2x+3}\)

Bài 1:

\(A=\frac{x^2+y^2-z^2+2xy}{x^2-y^2+z^2+2xz}\)

       \(=\frac{\left(x^2+2xy+y^2\right)-z^2}{\left(x^2+2xz+z^2\right)-y^2}\)

         \(=\frac{\left(x+y\right)^2-z^2}{\left(x+z\right)^2-y^2}\)

           \(=\frac{\left(x+y+z\right)\left(x+y+z\right)}{\left(x+y+z\right)\left(x-y+z\right)}\)

               \(=\frac{x+y-z}{x-y+z}\)

Ta thay : \(x=0;y=2009;z=2010\) ta được :

\(A=\frac{0+2009-2010}{0-2009+2010}=-\frac{1}{1}=-1\)

Chúc bạn học tốt !!!

\(A=\frac{x^2+y^2-z^2+2xy}{x^2-y^2+z^2+2xz}=\frac{\left(x^2+2xy+y^2\right)-z^2}{\left(x^2+2xz+z^2\right)-y^2}=\frac{\left(x+y\right)^2-z^2}{\left(x+z\right)^2-y^2}\)

\(=\frac{\left(x+y+z\right)\left(x+y-z\right)}{\left(x+y+z\right)\left(x-y+z\right)}=\frac{x+y-z}{x-y+z}\)

Thay \(\hept{\begin{cases}x=0\\y=2009\\z=2010\end{cases}}\) vào biểu thức :

\(\Rightarrow A=\frac{0+2009-2010}{0-2009+2010}=-1\)