Solve 1/x^2-4x+7+x+1/x

$$\frac{ 1 }{ { x }^{ 2 } -4x+7 } +x+ \frac{ 1 }{ x }$$

Evaluate

$$\frac{x^{4}-4x^{3}+8x^{2}-3x+7}{x\left(x^{2}-4x+7\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{x^{2}-4x+7}{x^{2}-4x+7}.

$$\frac{1}{x^{2}-4x+7}+\frac{x\left(x^{2}-4x+7\right)}{x^{2}-4x+7}+\frac{1}{x}$$

Since \frac{1}{x^{2}-4x+7} and \frac{x\left(x^{2}-4x+7\right)}{x^{2}-4x+7} have the same denominator, add them by adding their numerators.

$$\frac{1+x\left(x^{2}-4x+7\right)}{x^{2}-4x+7}+\frac{1}{x}$$

Do the multiplications in 1+x\left(x^{2}-4x+7\right).

$$\frac{1+x^{3}-4x^{2}+7x}{x^{2}-4x+7}+\frac{1}{x}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x^{2}-4x+7 and x is x\left(x^{2}-4x+7\right). Multiply \frac{1+x^{3}-4x^{2}+7x}{x^{2}-4x+7} times \frac{x}{x}. Multiply \frac{1}{x} times \frac{x^{2}-4x+7}{x^{2}-4x+7}.

$$\frac{\left(1+x^{3}-4x^{2}+7x\right)x}{x\left(x^{2}-4x+7\right)}+\frac{x^{2}-4x+7}{x\left(x^{2}-4x+7\right)}$$

Since \frac{\left(1+x^{3}-4x^{2}+7x\right)x}{x\left(x^{2}-4x+7\right)} and \frac{x^{2}-4x+7}{x\left(x^{2}-4x+7\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(1+x^{3}-4x^{2}+7x\right)x+x^{2}-4x+7}{x\left(x^{2}-4x+7\right)}$$

Do the multiplications in \left(1+x^{3}-4x^{2}+7x\right)x+x^{2}-4x+7.

$$\frac{x+x^{4}-4x^{3}+7x^{2}+x^{2}-4x+7}{x\left(x^{2}-4x+7\right)}$$

Combine like terms in x+x^{4}-4x^{3}+7x^{2}+x^{2}-4x+7.

$$\frac{-3x+x^{4}-4x^{3}+8x^{2}+7}{x\left(x^{2}-4x+7\right)}$$

Expand x\left(x^{2}-4x+7\right).

$$\frac{-3x+x^{4}-4x^{3}+8x^{2}+7}{x^{3}-4x^{2}+7x}$$

Differentiate w.r.t. x

$$\frac{x^{6}-8x^{5}+29x^{4}-50x^{3}+23x^{2}+56x-49}{\left(x\left(x^{2}-4x+7\right)\right)^{2}}$$