Solve sumfrac{x^{3}}{x^{5}-x^3-x^2+1}+1/x^2+x+1=2/x^2-1

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

Solve for Σ

$$Σ=\frac{x^{3}+x^{2}+x-3}{x^{3}}$$

$$x\neq 0\text{ and }|x|\neq 1$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x+1\right)\left(x-1\right)^{2}\left(x^{2}+x+1\right), the least common multiple of x^{5}-x^{3}-x^{2}+1,x^{2}+x+1,x^{2}-1.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

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

Use the distributive property to multiply x+1 by x^{2}-2x+1 and combine like terms.

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

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

$$Σx^{3}+x^{3}-x^{2}-x+1=2x^{3}-2$$

Subtract x^{3} from both sides.

$$Σx^{3}-x^{2}-x+1=2x^{3}-2-x^{3}$$

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

$$Σx^{3}-x^{2}-x+1=x^{3}-2$$

Add x^{2} to both sides.

$$Σx^{3}-x+1=x^{3}-2+x^{2}$$

Add x to both sides.

$$Σx^{3}+1=x^{3}-2+x^{2}+x$$

Subtract 1 from both sides.

$$Σx^{3}=x^{3}-2+x^{2}+x-1$$

Subtract 1 from -2 to get -3.

$$Σx^{3}=x^{3}-3+x^{2}+x$$

The equation is in standard form.

$$x^{3}Σ=x^{3}+x^{2}+x-3$$

Divide both sides by x^{3}.

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

Dividing by x^{3} undoes the multiplication by x^{3}.

$$Σ=\frac{x^{3}+x^{2}+x-3}{x^{3}}$$