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

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

Evaluate

$$-\frac{6x^{3}+13x^{2}+9x+1}{\left(1-x^{2}\right)\left(x^{2}+x+1\right)}$$

Short Solution Steps

Factor x^{2}-1.

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

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

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

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

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

Combine like terms in x+1+2x+1.

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

Factor 1-x^{3}.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-1\right)\left(x+1\right) and \left(x-1\right)\left(-x^{2}-x-1\right) is \left(x-1\right)\left(x+1\right)\left(-x^{2}-x-1\right). Multiply \frac{3x+2}{\left(x-1\right)\left(x+1\right)} times \frac{-x^{2}-x-1}{-x^{2}-x-1}. Multiply \frac{3x^{2}+5x-1}{\left(x-1\right)\left(-x^{2}-x-1\right)} times \frac{x+1}{x+1}.

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

Since \frac{\left(3x+2\right)\left(-x^{2}-x-1\right)}{\left(x-1\right)\left(x+1\right)\left(-x^{2}-x-1\right)} and \frac{\left(3x^{2}+5x-1\right)\left(x+1\right)}{\left(x-1\right)\left(x+1\right)\left(-x^{2}-x-1\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(3x+2\right)\left(-x^{2}-x-1\right)-\left(3x^{2}+5x-1\right)\left(x+1\right).

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

Combine like terms in -3x^{3}-3x^{2}-3x-2x^{2}-2x-2-3x^{3}-3x^{2}-5x^{2}-5x+x+1.

$$\frac{-6x^{3}-13x^{2}-9x-1}{\left(x-1\right)\left(x+1\right)\left(-x^{2}-x-1\right)}$$

Expand \left(x-1\right)\left(x+1\right)\left(-x^{2}-x-1\right).

$$\frac{-6x^{3}-13x^{2}-9x-1}{-x^{4}-x^{3}+x+1}$$

Expand

$$-\frac{6x^{3}+13x^{2}+9x+1}{\left(x+1\right)\left(1-x^{3}\right)}$$

Short Solution Steps

Factor x^{2}-1.

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

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

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

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

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

Combine like terms in x+1+2x+1.

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

Factor 1-x^{3}.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-1\right)\left(x+1\right) and \left(x-1\right)\left(-x^{2}-x-1\right) is \left(x-1\right)\left(x+1\right)\left(-x^{2}-x-1\right). Multiply \frac{3x+2}{\left(x-1\right)\left(x+1\right)} times \frac{-x^{2}-x-1}{-x^{2}-x-1}. Multiply \frac{3x^{2}+5x-1}{\left(x-1\right)\left(-x^{2}-x-1\right)} times \frac{x+1}{x+1}.

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

Since \frac{\left(3x+2\right)\left(-x^{2}-x-1\right)}{\left(x-1\right)\left(x+1\right)\left(-x^{2}-x-1\right)} and \frac{\left(3x^{2}+5x-1\right)\left(x+1\right)}{\left(x-1\right)\left(x+1\right)\left(-x^{2}-x-1\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(3x+2\right)\left(-x^{2}-x-1\right)-\left(3x^{2}+5x-1\right)\left(x+1\right).

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

Combine like terms in -3x^{3}-3x^{2}-3x-2x^{2}-2x-2-3x^{3}-3x^{2}-5x^{2}-5x+x+1.

$$\frac{-6x^{3}-13x^{2}-9x-1}{\left(x-1\right)\left(x+1\right)\left(-x^{2}-x-1\right)}$$

Expand \left(x-1\right)\left(x+1\right)\left(-x^{2}-x-1\right).

$$\frac{-6x^{3}-13x^{2}-9x-1}{-x^{4}-x^{3}+x+1}$$