Solve 1/x^3+x-72+1/x^2+2x+1

$$\frac { 1 } { x ^ { 3 } + x - 72 } + \frac { 1 } { x ^ { 2 } + 2 x + 1 }$$

Evaluate

$$\frac{x^{3}+x^{2}+3x-71}{\left(x+1\right)^{2}\left(x^{3}+x-72\right)}$$

Short Solution Steps

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

$$\frac{1}{x^{3}+x-72}+\frac{1}{\left(x+1\right)^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x^{3}+x-72 and \left(x+1\right)^{2} is \left(x+1\right)^{2}\left(x^{3}+x-72\right). Multiply \frac{1}{x^{3}+x-72} times \frac{\left(x+1\right)^{2}}{\left(x+1\right)^{2}}. Multiply \frac{1}{\left(x+1\right)^{2}} times \frac{x^{3}+x-72}{x^{3}+x-72}.

$$\frac{\left(x+1\right)^{2}}{\left(x+1\right)^{2}\left(x^{3}+x-72\right)}+\frac{x^{3}+x-72}{\left(x+1\right)^{2}\left(x^{3}+x-72\right)}$$

Since \frac{\left(x+1\right)^{2}}{\left(x+1\right)^{2}\left(x^{3}+x-72\right)} and \frac{x^{3}+x-72}{\left(x+1\right)^{2}\left(x^{3}+x-72\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(x+1\right)^{2}+x^{3}+x-72}{\left(x+1\right)^{2}\left(x^{3}+x-72\right)}$$

Do the multiplications in \left(x+1\right)^{2}+x^{3}+x-72.

$$\frac{x^{2}+2x+1+x^{3}+x-72}{\left(x+1\right)^{2}\left(x^{3}+x-72\right)}$$

Combine like terms in x^{2}+2x+1+x^{3}+x-72.

$$\frac{x^{2}+3x-71+x^{3}}{\left(x+1\right)^{2}\left(x^{3}+x-72\right)}$$

Expand \left(x+1\right)^{2}\left(x^{3}+x-72\right).

$$\frac{x^{2}+3x-71+x^{3}}{x^{5}+2x^{4}+2x^{3}-70x^{2}-143x-72}$$

Differentiate w.r.t. x

$$\frac{-2x^{6}-3x^{5}-13x^{4}+278x^{3}-8x^{2}+285x-10369}{\left(x+1\right)^{3}\left(x^{6}+2x^{4}-144x^{3}+x^{2}-144x+5184\right)}$$