Solve z^3+2left(sqrt{2}-1right)z^2+4left(1-sqrt{2}right)z-8=left(z^2-2right)left(z+xz+yright)

$${ z }^{ 3 } +2 \left( \sqrt{ 2 } -1 \right) { z }^{ 2 } +4 \left( 1- \sqrt{ 2 } \right) z-8 = \left( { z }^{ 2 } -2 \right) \left( z+xz+y \right)$$

Solve for x

$$\left\{\begin{matrix}x=-\frac{-yz^{2}+2\sqrt{2}z^{2}-2z^{2}-4\sqrt{2}z+2y+6z-8}{z\left(2-z^{2}\right)}\text{, }&z\neq 0\text{ and }|z|\neq \sqrt{2}\\x\in \mathrm{R}\text{, }&z=0\text{ and }y=4\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 2 by \sqrt{2}-1.

$$z^{3}+\left(2\sqrt{2}-2\right)z^{2}+4\left(1-\sqrt{2}\right)z-8=\left(z^{2}-2\right)\left(z+xz+y\right)$$

Use the distributive property to multiply 2\sqrt{2}-2 by z^{2}.

$$z^{3}+2\sqrt{2}z^{2}-2z^{2}+4\left(1-\sqrt{2}\right)z-8=\left(z^{2}-2\right)\left(z+xz+y\right)$$

Use the distributive property to multiply 4 by 1-\sqrt{2}.

$$z^{3}+2\sqrt{2}z^{2}-2z^{2}+\left(4-4\sqrt{2}\right)z-8=\left(z^{2}-2\right)\left(z+xz+y\right)$$

Use the distributive property to multiply 4-4\sqrt{2} by z.

$$z^{3}+2\sqrt{2}z^{2}-2z^{2}+4z-4\sqrt{2}z-8=\left(z^{2}-2\right)\left(z+xz+y\right)$$

Use the distributive property to multiply z^{2}-2 by z+xz+y.

$$z^{3}+2\sqrt{2}z^{2}-2z^{2}+4z-4\sqrt{2}z-8=z^{3}+xz^{3}+z^{2}y-2z-2xz-2y$$

Swap sides so that all variable terms are on the left hand side.

$$z^{3}+xz^{3}+z^{2}y-2z-2xz-2y=z^{3}+2\sqrt{2}z^{2}-2z^{2}+4z-4\sqrt{2}z-8$$

Subtract z^{3} from both sides.

$$xz^{3}+z^{2}y-2z-2xz-2y=z^{3}+2\sqrt{2}z^{2}-2z^{2}+4z-4\sqrt{2}z-8-z^{3}$$

Combine z^{3} and -z^{3} to get 0.

$$xz^{3}+z^{2}y-2z-2xz-2y=2\sqrt{2}z^{2}-2z^{2}+4z-4\sqrt{2}z-8$$

Subtract z^{2}y from both sides.

$$xz^{3}-2z-2xz-2y=2\sqrt{2}z^{2}-2z^{2}+4z-4\sqrt{2}z-8-z^{2}y$$

Add 2z to both sides.

$$xz^{3}-2xz-2y=2\sqrt{2}z^{2}-2z^{2}+4z-4\sqrt{2}z-8-z^{2}y+2z$$

Add 2y to both sides.

$$xz^{3}-2xz=2\sqrt{2}z^{2}-2z^{2}+4z-4\sqrt{2}z-8-z^{2}y+2z+2y$$

Combine 4z and 2z to get 6z.

$$xz^{3}-2xz=2\sqrt{2}z^{2}-2z^{2}+6z-4\sqrt{2}z-8-z^{2}y+2y$$

Combine all terms containing x.

$$\left(z^{3}-2z\right)x=2\sqrt{2}z^{2}-2z^{2}+6z-4\sqrt{2}z-8-z^{2}y+2y$$

The equation is in standard form.

$$\left(z^{3}-2z\right)x=-yz^{2}+2\sqrt{2}z^{2}-2z^{2}-4\sqrt{2}z+2y+6z-8$$

Divide both sides by z^{3}-2z.

$$\frac{\left(z^{3}-2z\right)x}{z^{3}-2z}=\frac{-yz^{2}+2\sqrt{2}z^{2}-2z^{2}-4\sqrt{2}z+2y+6z-8}{z^{3}-2z}$$

Dividing by z^{3}-2z undoes the multiplication by z^{3}-2z.

$$x=\frac{-yz^{2}+2\sqrt{2}z^{2}-2z^{2}-4\sqrt{2}z+2y+6z-8}{z^{3}-2z}$$

Divide 2z^{2}\sqrt{2}-2z^{2}+6z-4\sqrt{2}z-8-z^{2}y+2y by z^{3}-2z.

$$x=\frac{-yz^{2}+2\sqrt{2}z^{2}-2z^{2}-4\sqrt{2}z+2y+6z-8}{z\left(z^{2}-2\right)}$$

Solve for y

$$y=-\frac{-xz^{3}+2\sqrt{2}z^{2}+2xz-2z^{2}-4\sqrt{2}z+6z-8}{2-z^{2}}$$

$$|z|\neq \sqrt{2}$$

Steps for Solving Linear Equation

Use the distributive property to multiply 2 by \sqrt{2}-1.

$$z^{3}+\left(2\sqrt{2}-2\right)z^{2}+4\left(1-\sqrt{2}\right)z-8=\left(z^{2}-2\right)\left(z+xz+y\right)$$

Use the distributive property to multiply 2\sqrt{2}-2 by z^{2}.

$$z^{3}+2\sqrt{2}z^{2}-2z^{2}+4\left(1-\sqrt{2}\right)z-8=\left(z^{2}-2\right)\left(z+xz+y\right)$$

Use the distributive property to multiply 4 by 1-\sqrt{2}.

$$z^{3}+2\sqrt{2}z^{2}-2z^{2}+\left(4-4\sqrt{2}\right)z-8=\left(z^{2}-2\right)\left(z+xz+y\right)$$

Use the distributive property to multiply 4-4\sqrt{2} by z.

$$z^{3}+2\sqrt{2}z^{2}-2z^{2}+4z-4\sqrt{2}z-8=\left(z^{2}-2\right)\left(z+xz+y\right)$$

Use the distributive property to multiply z^{2}-2 by z+xz+y.

$$z^{3}+2\sqrt{2}z^{2}-2z^{2}+4z-4\sqrt{2}z-8=z^{3}+xz^{3}+z^{2}y-2z-2xz-2y$$

Swap sides so that all variable terms are on the left hand side.

$$z^{3}+xz^{3}+z^{2}y-2z-2xz-2y=z^{3}+2\sqrt{2}z^{2}-2z^{2}+4z-4\sqrt{2}z-8$$

Subtract z^{3} from both sides.

$$xz^{3}+z^{2}y-2z-2xz-2y=z^{3}+2\sqrt{2}z^{2}-2z^{2}+4z-4\sqrt{2}z-8-z^{3}$$

Combine z^{3} and -z^{3} to get 0.

$$xz^{3}+z^{2}y-2z-2xz-2y=2\sqrt{2}z^{2}-2z^{2}+4z-4\sqrt{2}z-8$$

Subtract xz^{3} from both sides.

$$z^{2}y-2z-2xz-2y=2\sqrt{2}z^{2}-2z^{2}+4z-4\sqrt{2}z-8-xz^{3}$$

Add 2z to both sides.

$$z^{2}y-2xz-2y=2\sqrt{2}z^{2}-2z^{2}+4z-4\sqrt{2}z-8-xz^{3}+2z$$

Add 2xz to both sides.

$$z^{2}y-2y=2\sqrt{2}z^{2}-2z^{2}+4z-4\sqrt{2}z-8-xz^{3}+2z+2xz$$

Combine 4z and 2z to get 6z.

$$z^{2}y-2y=2\sqrt{2}z^{2}-2z^{2}+6z-4\sqrt{2}z-8-xz^{3}+2xz$$

Combine all terms containing y.

$$\left(z^{2}-2\right)y=2\sqrt{2}z^{2}-2z^{2}+6z-4\sqrt{2}z-8-xz^{3}+2xz$$

The equation is in standard form.

$$\left(z^{2}-2\right)y=-xz^{3}+2\sqrt{2}z^{2}+2xz-2z^{2}-4\sqrt{2}z+6z-8$$

Divide both sides by z^{2}-2.

$$\frac{\left(z^{2}-2\right)y}{z^{2}-2}=\frac{-xz^{3}+2\sqrt{2}z^{2}+2xz-2z^{2}-4\sqrt{2}z+6z-8}{z^{2}-2}$$

Dividing by z^{2}-2 undoes the multiplication by z^{2}-2.

$$y=\frac{-xz^{3}+2\sqrt{2}z^{2}+2xz-2z^{2}-4\sqrt{2}z+6z-8}{z^{2}-2}$$