Solve 5x^3-8x^2+x-1/x^3-x

$$5 { x }^{ 3 } -8 { x }^{ 2 } +x- \frac{ 1 }{ { x }^{ 3 } -x }$$

Evaluate

$$\frac{5x^{6}-8x^{5}-4x^{4}+8x^{3}-x^{2}-1}{x\left(x^{2}-1\right)}$$

Short Solution Steps

Factor x^{3}-x.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 5x^{3}-8x^{2}+x times \frac{x\left(x-1\right)\left(x+1\right)}{x\left(x-1\right)\left(x+1\right)}.

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

Since \frac{\left(5x^{3}-8x^{2}+x\right)x\left(x-1\right)\left(x+1\right)}{x\left(x-1\right)\left(x+1\right)} and \frac{1}{x\left(x-1\right)\left(x+1\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(5x^{3}-8x^{2}+x\right)x\left(x-1\right)\left(x+1\right)-1.

$$\frac{5x^{6}-5x^{4}-8x^{5}+8x^{3}+x^{4}-x^{2}-1}{x\left(x-1\right)\left(x+1\right)}$$

Combine like terms in 5x^{6}-5x^{4}-8x^{5}+8x^{3}+x^{4}-x^{2}-1.

$$\frac{5x^{6}-4x^{4}-8x^{5}+8x^{3}-x^{2}-1}{x\left(x-1\right)\left(x+1\right)}$$

Expand x\left(x-1\right)\left(x+1\right).

$$\frac{5x^{6}-4x^{4}-8x^{5}+8x^{3}-x^{2}-1}{x^{3}-x}$$

Differentiate w.r.t. x

$$\frac{15x^{8}-16x^{7}-29x^{6}+32x^{5}+13x^{4}-16x^{3}+4x^{2}-1}{\left(x\left(x^{2}-1\right)\right)^{2}}$$