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[y+z x y]

[z+x z x]

[x+y y z]

=(x +y+z) (x-z)²

$Δ = | \begin{matrix} y + z & x & y \\ z + x & z & x \\ x + y & y & z \end{matrix} | = (y + z) | \begin{matrix} z & x \\ y & z \end{matrix} | - x | \begin{matrix} z + x & x \\ x + y & z \end{matrix} | + y | \begin{matrix} z + x & z \\ x + y & y \end{matrix} | = (y + z) (z^{2} - x y) - x [z (z + x) - x (x + y)] + y [y (z + x) - z (x + y)] = (y + z) (z^{2} - x y) - x [z^{2} + z x - x^{2} - x y] + y [y z + y x - z x - z y] = y z^{2} - x y^{2} + z^{3} - x y z - x z^{2} - x^{2} z + x^{3} + x^{2} y + y^{2} z + y^{2} x - x y z - y^{2} z = y z^{2} + z^{3} - 2 x y z - x z^{2} - x^{2} z + x^{3} + x^{2} y = (x + y + z) {(x - z)}^{2}$

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