Solve x^2+y^2+z^2+2xy+2yz+2zx-2x-2y-2z+1

$$x ^ { 2 } + y ^ { 2 } + z ^ { 2 } + 2 x y + 2 y z + 2 z x - 2 x - 2 y - 2 z + 1$$

Factor

$$\left(x+y+z-1\right)^{2}$$

Solution Steps

Consider x^{2}+y^{2}+z^{2}+2xy+2yz+2zx-2x-2y-2z+1 as a polynomial over variable x.

$$x^{2}+\left(2y+2z-2\right)x+y^{2}+z^{2}+2yz-2y-2z+1$$

Find one factor of the form x^{k}+m, where x^{k} divides the monomial with the highest power x^{2} and m divides the constant factor y^{2}+2yz-2y+z^{2}-2z+1. One such factor is x+y+z-1. Factor the polynomial by dividing it by this factor.

$$\left(x+y+z-1\right)\left(x+y+z-1\right)$$

Evaluate

$$\left(x+y+z-1\right)^{2}$$