Solve -x^3+2x^2-x+5/x-1=5/x-1+x-x^2

$$\frac { - x ^ { 3 } + 2 x ^ { 2 } - x + 5 } { x - 1 } = \frac { 5 } { x - 1 } + x - x ^ { 2 }$$

Solve for x

$$x\neq 1$$

Solution Steps

Variable x cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by x-1.

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

Use the distributive property to multiply x-1 by x.

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

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

$$-x^{3}+2x^{2}-x+5=5+x^{2}-x-x^{3}+x^{2}$$

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

$$-x^{3}+2x^{2}-x+5=5+2x^{2}-x-x^{3}$$

Subtract 5 from both sides.

$$-x^{3}+2x^{2}-x+5-5=2x^{2}-x-x^{3}$$

Subtract 5 from 5 to get 0.

$$-x^{3}+2x^{2}-x=2x^{2}-x-x^{3}$$

Subtract 2x^{2} from both sides.

$$-x^{3}+2x^{2}-x-2x^{2}=-x-x^{3}$$

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

$$-x^{3}-x=-x-x^{3}$$

Add x to both sides.

$$-x^{3}-x+x=-x^{3}$$

Combine -x and x to get 0.

$$-x^{3}=-x^{3}$$

Add x^{3} to both sides.

$$-x^{3}+x^{3}=0$$

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

$$0=0$$

Compare 0 and 0.

$$\text{true}$$

This is true for any x.

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

Variable x cannot be equal to 1.

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