Solve 1-u^2-2u/u^3+u^2+u+1=1/u+1-2u/u^2+1

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

Solve for u (complex solution)

$$u\in \mathrm{C}\setminus i,-i,-1$$

Solution Steps

Variable u cannot be equal to any of the values -1,-i,i since division by zero is not defined. Multiply both sides of the equation by \left(u+1\right)\left(u-i\right)\left(u+i\right), the least common multiple of u^{3}+u^{2}+u+1,u+1,u^{2}+1.

$$1-u^{2}-2u=u^{2}+1-\left(u+1\right)\times 2u$$

Use the distributive property to multiply u+1 by 2.

$$1-u^{2}-2u=u^{2}+1-\left(2u+2\right)u$$

Use the distributive property to multiply 2u+2 by u.

$$1-u^{2}-2u=u^{2}+1-\left(2u^{2}+2u\right)$$

To find the opposite of 2u^{2}+2u, find the opposite of each term.

$$1-u^{2}-2u=u^{2}+1-2u^{2}-2u$$

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

$$1-u^{2}-2u=-u^{2}+1-2u$$

Add u^{2} to both sides.

$$1-u^{2}-2u+u^{2}=1-2u$$

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

$$1-2u=1-2u$$

Subtract 1 from both sides.

$$1-2u-1=-2u$$

Subtract 1 from 1 to get 0.

$$-2u=-2u$$

Add 2u to both sides.

$$-2u+2u=0$$

Combine -2u and 2u to get 0.

$$0=0$$

Compare 0 and 0.

$$\text{true}$$

This is true for any u.

$$u\in \mathrm{C}$$

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

$$u\in \mathrm{C}\setminus -i,i,-1$$

Solve for u

$$u\neq -1$$

Solution Steps

Variable u cannot be equal to -1 since division by zero is not defined. Multiply both sides of the equation by \left(u+1\right)\left(u^{2}+1\right), the least common multiple of u^{3}+u^{2}+u+1,u+1,u^{2}+1.

$$1-u^{2}-2u=u^{2}+1-\left(u+1\right)\times 2u$$

Use the distributive property to multiply u+1 by 2.

$$1-u^{2}-2u=u^{2}+1-\left(2u+2\right)u$$

Use the distributive property to multiply 2u+2 by u.

$$1-u^{2}-2u=u^{2}+1-\left(2u^{2}+2u\right)$$

To find the opposite of 2u^{2}+2u, find the opposite of each term.

$$1-u^{2}-2u=u^{2}+1-2u^{2}-2u$$

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

$$1-u^{2}-2u=-u^{2}+1-2u$$

Add u^{2} to both sides.

$$1-u^{2}-2u+u^{2}=1-2u$$

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

$$1-2u=1-2u$$

Subtract 1 from both sides.

$$1-2u-1=-2u$$

Subtract 1 from 1 to get 0.

$$-2u=-2u$$

Add 2u to both sides.

$$-2u+2u=0$$

Combine -2u and 2u to get 0.

$$0=0$$

Compare 0 and 0.

$$\text{true}$$

This is true for any u.

$$u\in \mathrm{R}$$

Variable u cannot be equal to -1.

$$u\in \mathrm{R}\setminus -1$$