Solve x-1/x^2-1+2/x^2+1+4/x^4+1+8/x^8-1=frac{1}{x-1}

$$\frac { x - 1 } { x ^ { 2 } - 1 } + \frac { 2 } { x ^ { 2 } + 1 } + \frac { 4 } { x ^ { 4 } + 1 } + \frac { 8 } { x ^ { 8 } - 1 } = \frac { 1 } { x - 1 }$$

Solve for x

$$x\in \mathrm{R}\setminus 1,-1$$

Solution Steps

Variable x cannot be equal to any of the values -1,1 since division by zero is not defined. Multiply both sides of the equation by \left(x-1\right)\left(x+1\right)\left(x^{2}+1\right)\left(x^{4}+1\right), the least common multiple of x^{2}-1,x^{2}+1,x^{4}+1,x^{8}-1,x-1.

$$\left(x^{2}+1\right)\left(x^{4}+1\right)\left(x-1\right)+\left(x^{4}+1\right)\left(x^{2}-1\right)\times 2+\left(x^{4}-1\right)\times 4+8=\left(x+1\right)\left(x^{2}+1\right)\left(x^{4}+1\right)$$

Use the distributive property to multiply x^{2}+1 by x^{4}+1.

$$\left(x^{6}+x^{2}+x^{4}+1\right)\left(x-1\right)+\left(x^{4}+1\right)\left(x^{2}-1\right)\times 2+\left(x^{4}-1\right)\times 4+8=\left(x+1\right)\left(x^{2}+1\right)\left(x^{4}+1\right)$$

Use the distributive property to multiply x^{6}+x^{2}+x^{4}+1 by x-1.

$$x^{7}-x^{6}+x^{3}-x^{2}+x^{5}-x^{4}+x-1+\left(x^{4}+1\right)\left(x^{2}-1\right)\times 2+\left(x^{4}-1\right)\times 4+8=\left(x+1\right)\left(x^{2}+1\right)\left(x^{4}+1\right)$$

Use the distributive property to multiply x^{4}+1 by x^{2}-1.

$$x^{7}-x^{6}+x^{3}-x^{2}+x^{5}-x^{4}+x-1+\left(x^{6}-x^{4}+x^{2}-1\right)\times 2+\left(x^{4}-1\right)\times 4+8=\left(x+1\right)\left(x^{2}+1\right)\left(x^{4}+1\right)$$

Use the distributive property to multiply x^{6}-x^{4}+x^{2}-1 by 2.

$$x^{7}-x^{6}+x^{3}-x^{2}+x^{5}-x^{4}+x-1+2x^{6}-2x^{4}+2x^{2}-2+\left(x^{4}-1\right)\times 4+8=\left(x+1\right)\left(x^{2}+1\right)\left(x^{4}+1\right)$$

Combine -x^{6} and 2x^{6} to get x^{6}.

$$x^{7}+x^{6}+x^{3}-x^{2}+x^{5}-x^{4}+x-1-2x^{4}+2x^{2}-2+\left(x^{4}-1\right)\times 4+8=\left(x+1\right)\left(x^{2}+1\right)\left(x^{4}+1\right)$$

Combine -x^{4} and -2x^{4} to get -3x^{4}.

$$x^{7}+x^{6}+x^{3}-x^{2}+x^{5}-3x^{4}+x-1+2x^{2}-2+\left(x^{4}-1\right)\times 4+8=\left(x+1\right)\left(x^{2}+1\right)\left(x^{4}+1\right)$$

Combine -x^{2} and 2x^{2} to get x^{2}.

$$x^{7}+x^{6}+x^{3}+x^{2}+x^{5}-3x^{4}+x-1-2+\left(x^{4}-1\right)\times 4+8=\left(x+1\right)\left(x^{2}+1\right)\left(x^{4}+1\right)$$

Subtract 2 from -1 to get -3.

$$x^{7}+x^{6}+x^{3}+x^{2}+x^{5}-3x^{4}+x-3+\left(x^{4}-1\right)\times 4+8=\left(x+1\right)\left(x^{2}+1\right)\left(x^{4}+1\right)$$

Use the distributive property to multiply x^{4}-1 by 4.

$$x^{7}+x^{6}+x^{3}+x^{2}+x^{5}-3x^{4}+x-3+4x^{4}-4+8=\left(x+1\right)\left(x^{2}+1\right)\left(x^{4}+1\right)$$

Combine -3x^{4} and 4x^{4} to get x^{4}.

$$x^{7}+x^{6}+x^{3}+x^{2}+x^{5}+x^{4}+x-3-4+8=\left(x+1\right)\left(x^{2}+1\right)\left(x^{4}+1\right)$$

Subtract 4 from -3 to get -7.

$$x^{7}+x^{6}+x^{3}+x^{2}+x^{5}+x^{4}+x-7+8=\left(x+1\right)\left(x^{2}+1\right)\left(x^{4}+1\right)$$

Add -7 and 8 to get 1.

$$x^{7}+x^{6}+x^{3}+x^{2}+x^{5}+x^{4}+x+1=\left(x+1\right)\left(x^{2}+1\right)\left(x^{4}+1\right)$$

Use the distributive property to multiply x+1 by x^{2}+1.

$$x^{7}+x^{6}+x^{3}+x^{2}+x^{5}+x^{4}+x+1=\left(x^{3}+x+x^{2}+1\right)\left(x^{4}+1\right)$$

Use the distributive property to multiply x^{3}+x+x^{2}+1 by x^{4}+1.

$$x^{7}+x^{6}+x^{3}+x^{2}+x^{5}+x^{4}+x+1=x^{7}+x^{3}+x^{5}+x+x^{6}+x^{2}+x^{4}+1$$

Subtract x^{7} from both sides.

$$x^{7}+x^{6}+x^{3}+x^{2}+x^{5}+x^{4}+x+1-x^{7}=x^{3}+x^{5}+x+x^{6}+x^{2}+x^{4}+1$$

Combine x^{7} and -x^{7} to get 0.

$$x^{6}+x^{3}+x^{2}+x^{5}+x^{4}+x+1=x^{3}+x^{5}+x+x^{6}+x^{2}+x^{4}+1$$

Subtract x^{3} from both sides.

$$x^{6}+x^{3}+x^{2}+x^{5}+x^{4}+x+1-x^{3}=x^{5}+x+x^{6}+x^{2}+x^{4}+1$$

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

$$x^{6}+x^{2}+x^{5}+x^{4}+x+1=x^{5}+x+x^{6}+x^{2}+x^{4}+1$$

Subtract x^{5} from both sides.

$$x^{6}+x^{2}+x^{5}+x^{4}+x+1-x^{5}=x+x^{6}+x^{2}+x^{4}+1$$

Combine x^{5} and -x^{5} to get 0.

$$x^{6}+x^{2}+x^{4}+x+1=x+x^{6}+x^{2}+x^{4}+1$$

Subtract x from both sides.

$$x^{6}+x^{2}+x^{4}+x+1-x=x^{6}+x^{2}+x^{4}+1$$

Combine x and -x to get 0.

$$x^{6}+x^{2}+x^{4}+1=x^{6}+x^{2}+x^{4}+1$$

Subtract x^{6} from both sides.

$$x^{6}+x^{2}+x^{4}+1-x^{6}=x^{2}+x^{4}+1$$

Combine x^{6} and -x^{6} to get 0.

$$x^{2}+x^{4}+1=x^{2}+x^{4}+1$$

Subtract x^{2} from both sides.

$$x^{2}+x^{4}+1-x^{2}=x^{4}+1$$

Combine x^{2} and -x^{2} to get 0.

$$x^{4}+1=x^{4}+1$$

Subtract x^{4} from both sides.

$$x^{4}+1-x^{4}=1$$

Combine x^{4} and -x^{4} to get 0.

$$1=1$$

Compare 1 and 1.

$$\text{true}$$

This is true for any x.

$$x\in \mathrm{R}$$

Variable x cannot be equal to any of the values -1,1.

$$x\in \mathrm{R}\setminus -1,1$$

Solve for x (complex solution)

$$x\in \mathrm{C}\setminus \sqrt{2}\left(-\frac{1}{2}-\frac{1}{2}i\right),\sqrt{2}\left(\frac{1}{2}+\frac{1}{2}i\right),\sqrt{2}\left(-\frac{1}{2}+\frac{1}{2}i\right),\sqrt{2}\left(\frac{1}{2}-\frac{1}{2}i\right),-1,-i,i,1$$