a) \(4x^4+y^4\)
\(=\left(2x^2\right)^2+2.2x^2.y^2+\left(y^2\right)^2-2.2x^2.y^2\)
\(=\left(2x^2+y^2\right)^2-4x^2y^2\)
\(=\left(2x^2+y^2\right)^2-\left(2xy\right)^2\)
\(=\left(2x^2+y^2+2xy\right)\left(2x^2+y^2-2xy\right)\)
b) \(\left(x^2-3x-1\right)^2-12\left(x^2-3x-1\right)+27\)
\(=\left(x^2-3x-1\right)^2-2\left(x^2-3x-1\right).6+36-9\)
\(=\left(x^2-3x-1-6\right)^2-3^2\)
\(=\left(x^2-3x-7\right)^2-3^2\)
\(=\left(x^2-3x-7-3\right)\left(x^2-3x-7+3\right)\)
\(=\left(x^2-3x-10\right)\left(x^2-3x-4\right)\)
c) \(x^3-x^2-5x+125\)
\(=x^3+5x^2-6x^2-30x+25x+125\)
\(=x^2\left(x+5\right)-6x\left(x+5\right)+25\left(x+5\right)\)
\(=\left(x+5\right)\left(x^2-6x+25\right)\)
d) \(xy\left(x+y\right)+yz\left(y+z\right)+zx\left(z+x\right)+2xyz\)
\(=xy\left(x+y\right)+yz\left(y+z\right)+xyz+zx\left(z+x\right)+xyz\)
\(=xy\left(x+y\right)+yz\left(y+z+x\right)+zx\left(z+x+y\right)\)
\(=xy\left(x+y\right)+z\left(x+y+z\right)\left(y+x\right)\)
\(=\left(x+y\right)\left[xy+z\left(x+y+z\right)\right]\)
\(=\left(x+y\right)\left(xy+zx+yz+z^2\right)\)
\(=\left(x+y\right)\left[y\left(x+z\right)+z\left(x+z\right)\right]\)
\(=\left(x+y\right)\left(x+z\right)\left(y+z\right)\)
a) ta có : \(4x^4+y^4=4x^4+4x^2y^2+y^2-\left(2xy\right)^2\)
\(=\left(2x^2+y^2\right)^2-\left(2xy\right)^2=\left(2x^2+y^2-2xy\right)\left(2x^2+y^2+2xy\right)\)
b) ta có : \(\left(x^3-3x-1\right)^2-12\left(x^2-3x-1\right)+27\)
\(=\left(x^2-3x-1\right)^2-3\left(x^2-3x-1\right)-9\left(x^2-3x-1\right)+27\)
\(=\left(x^2-3x-1\right)\left(x^2-3x-4\right)-9\left(x^2-3x-4\right)\)
\(=\left(x^2-3x-10\right)\left(x^2-3x-4\right)\)
c) ta có : \(x^3-x^2-5x+125=x^2+5x^2-6x^2-30x+25x+125\)
\(=x^2\left(x+5\right)-6x\left(x+5\right)+25\left(x+5\right)=\left(x^2-6x+25\right)\left(x+5\right)\)
d) ta có : \(xy\left(x+y\right)+yz\left(y+z\right)+zx\left(z+x\right)+2xyz\)
\(=x^2y+xy^2+y^2z+xyz+yz^2+z^2x+zx^2+xyz\)
\(=y\left(x^2+xy+yz+xz\right)+z\left(yz+zx+x^2+xy\right)\)
\(=\left(x+y\right)\left(x^2+xy+yz+xz\right)\)
