Solve x^2+1/x+15-1/x-1

$$\frac { x ^ { 2 } + 1 } { x + 15 } - \frac { 1 } { x - 1 }$$

Evaluate

$$\frac{x^{3}-x^{2}-16}{\left(x-1\right)\left(x+15\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+15 and x-1 is \left(x-1\right)\left(x+15\right). Multiply \frac{x^{2}+1}{x+15} times \frac{x-1}{x-1}. Multiply \frac{1}{x-1} times \frac{x+15}{x+15}.

$$\frac{\left(x^{2}+1\right)\left(x-1\right)}{\left(x-1\right)\left(x+15\right)}-\frac{x+15}{\left(x-1\right)\left(x+15\right)}$$

Since \frac{\left(x^{2}+1\right)\left(x-1\right)}{\left(x-1\right)\left(x+15\right)} and \frac{x+15}{\left(x-1\right)\left(x+15\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x^{2}+1\right)\left(x-1\right)-\left(x+15\right)}{\left(x-1\right)\left(x+15\right)}$$

Do the multiplications in \left(x^{2}+1\right)\left(x-1\right)-\left(x+15\right).

$$\frac{x^{3}-x^{2}+x-1-x-15}{\left(x-1\right)\left(x+15\right)}$$

Combine like terms in x^{3}-x^{2}+x-1-x-15.

$$\frac{x^{3}-x^{2}-16}{\left(x-1\right)\left(x+15\right)}$$

Expand \left(x-1\right)\left(x+15\right).

$$\frac{x^{3}-x^{2}-16}{x^{2}+14x-15}$$

Expand

$$\frac{x^{3}-x^{2}-16}{\left(x-1\right)\left(x+15\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+15 and x-1 is \left(x-1\right)\left(x+15\right). Multiply \frac{x^{2}+1}{x+15} times \frac{x-1}{x-1}. Multiply \frac{1}{x-1} times \frac{x+15}{x+15}.

$$\frac{\left(x^{2}+1\right)\left(x-1\right)}{\left(x-1\right)\left(x+15\right)}-\frac{x+15}{\left(x-1\right)\left(x+15\right)}$$

Since \frac{\left(x^{2}+1\right)\left(x-1\right)}{\left(x-1\right)\left(x+15\right)} and \frac{x+15}{\left(x-1\right)\left(x+15\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x^{2}+1\right)\left(x-1\right)-\left(x+15\right)}{\left(x-1\right)\left(x+15\right)}$$

Do the multiplications in \left(x^{2}+1\right)\left(x-1\right)-\left(x+15\right).

$$\frac{x^{3}-x^{2}+x-1-x-15}{\left(x-1\right)\left(x+15\right)}$$

Combine like terms in x^{3}-x^{2}+x-1-x-15.

$$\frac{x^{3}-x^{2}-16}{\left(x-1\right)\left(x+15\right)}$$

Expand \left(x-1\right)\left(x+15\right).

$$\frac{x^{3}-x^{2}-16}{x^{2}+14x-15}$$