Solve g(x)=x+1+1/(x-1)^2

$$g ( x ) = x + 1 + \frac { 1 } { ( x - 1 ) ^ { 2 } }$$

Evaluate

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

Short Solution Steps

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

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

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

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

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

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

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

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

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

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

Expand

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

Short Solution Steps

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

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

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

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

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

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

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

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

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

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