Solve 5/x-9/x^2+x+4/x^3-x^2

$$\frac { 5 } { x } - \frac { 9 } { x ^ { 2 } + x } + \frac { 4 } { x ^ { 3 } - x ^ { 2 } }$$

Evaluate

$$\frac{5x^{3}-9x^{2}+8x+4}{x^{2}\left(x^{2}-1\right)}$$

Short Solution Steps

Factor x^{2}+x.

$$\frac{5}{x}-\frac{9}{x\left(x+1\right)}+\frac{4}{x^{3}-x^{2}}$$

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{5}{x} times \frac{x+1}{x+1}.

$$\frac{5\left(x+1\right)}{x\left(x+1\right)}-\frac{9}{x\left(x+1\right)}+\frac{4}{x^{3}-x^{2}}$$

Since \frac{5\left(x+1\right)}{x\left(x+1\right)} and \frac{9}{x\left(x+1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{5\left(x+1\right)-9}{x\left(x+1\right)}+\frac{4}{x^{3}-x^{2}}$$

Do the multiplications in 5\left(x+1\right)-9.

$$\frac{5x+5-9}{x\left(x+1\right)}+\frac{4}{x^{3}-x^{2}}$$

Combine like terms in 5x+5-9.

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

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

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

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{5x-4}{x\left(x+1\right)} times \frac{x\left(x-1\right)}{x\left(x-1\right)}. Multiply \frac{4}{\left(x-1\right)x^{2}} times \frac{x+1}{x+1}.

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

Since \frac{\left(5x-4\right)x\left(x-1\right)}{\left(x-1\right)\left(x+1\right)x^{2}} and \frac{4\left(x+1\right)}{\left(x-1\right)\left(x+1\right)x^{2}} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(5x-4\right)x\left(x-1\right)+4\left(x+1\right).

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

Combine like terms in 5x^{3}-5x^{2}-4x^{2}+4x+4x+4.

$$\frac{5x^{3}-9x^{2}+8x+4}{\left(x-1\right)\left(x+1\right)x^{2}}$$

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

$$\frac{5x^{3}-9x^{2}+8x+4}{x^{4}-x^{2}}$$

Differentiate w.r.t. x

$$\frac{8+8x-16x^{2}-29x^{3}+18x^{4}-5x^{5}}{x^{3}\left(x^{2}-1\right)^{2}}$$