Ôn tập phép nhân và phép chia đa thức

Bài 1:

a) \(x^4+64\)

\(=\left(x^2\right)^2+2.x^2.8+8^2-2.x^2.8\)

\(=\left(x^2+8\right)^2-\left(4x\right)^2\)

\(=\left(x^2+8-4x\right)\left(x^2+8+4x\right)\)

b) \(x^5+x^4+1\)

\(=x^5+x^4+x^3+x^2-x^3-x^2-x+x+1\)

\(=\left(x^5+x^4+x^3\right)-\left(x^3+x^2+x\right)+\left(x^2+x+1\right)\)

\(=x^3\left(x^2+x+1\right)-x\left(x^2+x+1\right)+\left(x^2+x+1\right)\)

\(=\left(x^2+x+1\right)\left(x^3-x+1\right)\)

c) \(x^3+y^3+z^3-3xyz\)

\(=\left(x+y\right)^3-3xy\left(x+y\right)+z^3-3xyz\)

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

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

\(=\left(x+y+z\right)\left[\left(x+y+z\right)^2-3xy-3yz-3xz\right]\)

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

\(=\left(x+y+z\right)\left(x^2+y^2+z^2-xy-xz-yz\right)\)

Bài 2:

\(xy+1=x+y\)

\(\Rightarrow xy+1-x-y=0\)

\(\Rightarrow\left(xy-x\right)-\left(y-1\right)=0\)

\(\Rightarrow x\left(y-1\right)-\left(y-1\right)=0\)

\(\Rightarrow\left(y-1\right)\left(x-1\right)=0\)

\(\Rightarrow\left\{{}\begin{matrix}y-1=0\\x-1=0\end{matrix}\right.\)

\(\Rightarrow\left\{{}\begin{matrix}y=1\\x=1\end{matrix}\right.\)