Solve 2p/+p^2+p^4-2p/1-p^2+p^4

$$\frac { 2 p } { + p ^ { 2 } + p ^ { 4 } } - \frac { 2 p } { 1 - p ^ { 2 } + p ^ { 4 } }$$

Evaluate

$$\frac{2\left(1-2p^{2}\right)}{p\left(p^{6}+1\right)}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{2p}{p^{2}+p^{4}}.

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

Cancel out p in both numerator and denominator.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of p\left(p^{2}+1\right) and 1-p^{2}+p^{4} is p\left(p^{2}+1\right)\left(p^{4}-p^{2}+1\right). Multiply \frac{2}{p\left(p^{2}+1\right)} times \frac{p^{4}-p^{2}+1}{p^{4}-p^{2}+1}. Multiply \frac{2p}{1-p^{2}+p^{4}} times \frac{p\left(p^{2}+1\right)}{p\left(p^{2}+1\right)}.

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

Since \frac{2\left(p^{4}-p^{2}+1\right)}{p\left(p^{2}+1\right)\left(p^{4}-p^{2}+1\right)} and \frac{2pp\left(p^{2}+1\right)}{p\left(p^{2}+1\right)\left(p^{4}-p^{2}+1\right)} have the same denominator, subtract them by subtracting their numerators.

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

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

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

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

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

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

$$\frac{-4p^{2}+2}{p^{7}+p}$$

Factor

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