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

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

Evaluate

$$\frac{x}{x^{2}-x+1}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x^{2}-x+1 and x+1 is \left(x+1\right)\left(x^{2}-x+1\right). Multiply \frac{x-2}{x^{2}-x+1} times \frac{x+1}{x+1}. Multiply \frac{1}{x+1} times \frac{x^{2}-x+1}{x^{2}-x+1}.

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

Since \frac{\left(x-2\right)\left(x+1\right)}{\left(x+1\right)\left(x^{2}-x+1\right)} and \frac{x^{2}-x+1}{\left(x+1\right)\left(x^{2}-x+1\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(x-2\right)\left(x+1\right)-\left(x^{2}-x+1\right).

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

Combine like terms in x^{2}+x-2x-2-x^{2}+x-1.

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

Factor x^{3}+1.

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

Since \frac{-3}{\left(x+1\right)\left(x^{2}-x+1\right)} and \frac{x^{2}+x+3}{\left(x+1\right)\left(x^{2}-x+1\right)} have the same denominator, add them by adding their numerators.

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

Combine like terms in -3+x^{2}+x+3.

$$\frac{x^{2}+x}{\left(x+1\right)\left(x^{2}-x+1\right)}$$

Factor the expressions that are not already factored in \frac{x^{2}+x}{\left(x+1\right)\left(x^{2}-x+1\right)}.

$$\frac{x\left(x+1\right)}{\left(x+1\right)\left(x^{2}-x+1\right)}$$

Cancel out x+1 in both numerator and denominator.

$$\frac{x}{x^{2}-x+1}$$

Expand

$$\frac{x}{x^{2}-x+1}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x^{2}-x+1 and x+1 is \left(x+1\right)\left(x^{2}-x+1\right). Multiply \frac{x-2}{x^{2}-x+1} times \frac{x+1}{x+1}. Multiply \frac{1}{x+1} times \frac{x^{2}-x+1}{x^{2}-x+1}.

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

Since \frac{\left(x-2\right)\left(x+1\right)}{\left(x+1\right)\left(x^{2}-x+1\right)} and \frac{x^{2}-x+1}{\left(x+1\right)\left(x^{2}-x+1\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(x-2\right)\left(x+1\right)-\left(x^{2}-x+1\right).

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

Combine like terms in x^{2}+x-2x-2-x^{2}+x-1.

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

Factor x^{3}+1.

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

Since \frac{-3}{\left(x+1\right)\left(x^{2}-x+1\right)} and \frac{x^{2}+x+3}{\left(x+1\right)\left(x^{2}-x+1\right)} have the same denominator, add them by adding their numerators.

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

Combine like terms in -3+x^{2}+x+3.

$$\frac{x^{2}+x}{\left(x+1\right)\left(x^{2}-x+1\right)}$$

Factor the expressions that are not already factored in \frac{x^{2}+x}{\left(x+1\right)\left(x^{2}-x+1\right)}.

$$\frac{x\left(x+1\right)}{\left(x+1\right)\left(x^{2}-x+1\right)}$$

Cancel out x+1 in both numerator and denominator.

$$\frac{x}{x^{2}-x+1}$$