a) \(16x^4\left(x-y\right)-x+y\)
\(=\left(4x^2\right)^2\left(x-y\right)-\left(x-y\right)\)
\(=\left(x-y\right)\left[\left(4x^2\right)^2-1\right]\)
\(=\left(x-y\right)\left(4x^2-1\right)\left(4x^2+1\right)\)
\(=\left(x-y\right)\left(4x^2+1\right)\left[\left(2x\right)^2-1\right]\)
\(=\left(x-y\right)\left(4x^2+1\right)\left(2x-1\right)\left(2x+1\right)\)
b) \(2x^3y-2xy^3-4xy^2-2xy\)
\(=2xy\left(x^2-y^2-2y-1\right)\)
\(=2xy\left[x^2-\left(y^2+2y+1\right)\right]\)
\(=2xy\left[x^2-\left(y+1\right)^2\right]\)
\(=2xy\left(x-y-1\right)\left(x+y+1\right)\)
c) \(x\left(y^2-z^2\right)+y\left(z^2-x^2\right)+z\left(x^2-y^2\right)\)
\(=x\left(y^2-z^2\right)-y\left[\left(y^2-z^2\right)+\left(x^2-y^2\right)\right]+z\left(x^2-y^2\right)\)
\(=x\left(y^2-z^2\right)-y\left(y^2-z^2\right)-y\left(x^2-y^2\right)+z\left(x^2-y^2\right)\)
\(=\left(y^2-z^2\right)\left(x-y\right)-\left(x^2-y^2\right)\left(y-z\right)\)
\(=\left(y-z\right)\left(y+z\right)\left(x-y\right)-\left(y-z\right)\left(x+y\right)\left(x-y\right)\)
\(=\left(y-z\right)\left(x-y\right)\left(y+z-x-y\right)\)
\(=\left(y-z\right)\left(x-y\right)\left(z-x\right)\)
