Solve 4/1-x^4+4/1+x^4

$$\frac { 4 } { 1 - x ^ { 4 } } + \frac { 4 } { 1 + x ^ { 4 } }$$

Evaluate

$$\frac{8}{1-x^{8}}$$

Short Solution Steps

Factor 1-x^{4}.

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

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

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

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

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

Do the multiplications in 4\left(x^{4}+1\right)+4\left(x-1\right)\left(x+1\right)\left(-x^{2}-1\right).

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

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

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

Expand \left(x-1\right)\left(x+1\right)\left(-x^{2}-1\right)\left(x^{4}+1\right).

$$\frac{8}{-x^{8}+1}$$

Factor

$$\frac{8}{1-x^{8}}$$