Luyện tập chung

a)

 \(x^2-6 x+9-y^2\)

\(=\left(x^2-6 x+9\right)-y^2\)

\(=(x-3)^2-y^2 \)

\(=(x-3+y)(x-3-y)\)
b) \(4 x^2-y^2+4 y-4\)

\(=(2 x)^2-\left(y^2-4 y+4\right)\)
\(=(2 x)^2-(y-2)^2\)

\(=(2 x-y+2)(2 x+y-2) ;\)
\(c) x y+z^2+x z+y z\)

\(=(x y+x z)+\left(z^2+y z\right)\)

\(=x(y+z) \\+z(z+y)\)

\(=(y+z)(x+z)\)
\(d)  x^2-4 x y+4 y^2+x z-2 y z\)

\(=\left(x^2-4 x y+4 y^2\right)+(x z-2 y z) \\\)

\(=(x-2 y)^2+z(x-2 y) \)

\(=(x-2 y)(x-2 y+z)\)

a) 

\( x^3+y^3+x+y\)

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

\(=(x+y)\left(x^2-x y+y^2+1\right)\)

b)

\(x^3-y^3+x-y\)

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

\(=(x-y)\left(x^2+x y+y^2+1\right)\)

c) 

\((x-y)^3+(x+y)^3 \\\)

\( =(x-y+x+y)\left[(x-y)^2-(x-y)(x+y)+(x+y)^2\right] \\ \)

\(=2 x \cdot\left(x^2-2 x y+y^2-x^2+y^2+x^2+2 x y+y^2\right) \\ \)

\(=2 x\left(x^2+3 y^2\right) \)

d) 

\(x^3-3 x^2 y+3 x y^2-y^3+y^2-x^2\)

\(=\left(x^3-3 x^2 y+3 x y^2-y^3\right) \\ +\left(y^2-x^2\right)\)

\(=(x-y)^3+(y-x)(y+x)\)

\(=(x-y)\left[(x-y)^2-y-x\right] \)

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