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

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

Evaluate

$$\frac{x^{3}-2x^{2}+4x+3}{x^{2}\left(x^{2}-1\right)}$$

Short Solution Steps

Factor x^{2}+x.

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

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

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

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

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

Combine like terms in x+1-2.

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

Factor x^{3}-x^{2}.

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

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

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

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

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

Do the multiplications in \left(x-1\right)x\left(x-1\right)+3\left(x+1\right).

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

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

$$\frac{x^{3}-2x^{2}+4x+3}{\left(x-1\right)\left(x+1\right)x^{2}}$$

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

$$\frac{x^{3}-2x^{2}+4x+3}{x^{4}-x^{2}}$$

Differentiate w.r.t. x

$$\frac{6+4x-12x^{2}-13x^{3}+4x^{4}-x^{5}}{x^{3}\left(x^{2}-1\right)^{2}}$$