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

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

Evaluate

$$\frac{1}{x^{4}+x^{2}+1}$$

Short Solution Steps

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

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

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

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

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

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

Combine like terms in x^{2}-x+1-x^{2}-x-1.

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

Factor x^{4}+x^{2}+1.

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

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

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

Combine like terms in -2x+2x+1.

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

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

$$\frac{1}{x^{4}+x^{2}+1}$$

Expand

$$\frac{1}{x^{4}+x^{2}+1}$$

Short Solution Steps

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

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

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

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

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

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

Combine like terms in x^{2}-x+1-x^{2}-x-1.

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

Factor x^{4}+x^{2}+1.

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

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

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

Combine like terms in -2x+2x+1.

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

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

$$\frac{1}{x^{4}+x^{2}+1}$$