Solve 1/x-1/x(x+1)1/(x+1)(x+1)

$$\frac{ 1 }{ x } - \frac{ 1 }{ x(x+1) } \frac{ 1 }{ (x+1)(x+1) }$$

Evaluate

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

Short Solution Steps

Multiply x+1 and x+1 to get \left(x+1\right)^{2}.

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

Use the distributive property to multiply x by x+1.

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

Multiply \frac{1}{x^{2}+x} times \frac{1}{\left(x+1\right)^{2}} by multiplying numerator times numerator and denominator times denominator.

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

Use the distributive property to multiply x^{2}+x by \left(x+1\right)^{2}.

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

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

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

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

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

Since \frac{\left(x+1\right)^{3}}{x\left(x+1\right)^{3}} and \frac{1}{x\left(x+1\right)^{3}} have the same denominator, subtract them by subtracting their numerators.

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

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

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

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

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

Factor the expressions that are not already factored in \frac{x^{3}+3x^{2}+3x}{x\left(x+1\right)^{3}}.

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

Cancel out x in both numerator and denominator.

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

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

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

Expand

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

Short Solution Steps

Multiply x+1 and x+1 to get \left(x+1\right)^{2}.

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

Use the distributive property to multiply x by x+1.

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

Multiply \frac{1}{x^{2}+x} times \frac{1}{\left(x+1\right)^{2}} by multiplying numerator times numerator and denominator times denominator.

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

Use the distributive property to multiply x^{2}+x by \left(x+1\right)^{2}.

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

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

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

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

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

Since \frac{\left(x+1\right)^{3}}{x\left(x+1\right)^{3}} and \frac{1}{x\left(x+1\right)^{3}} have the same denominator, subtract them by subtracting their numerators.

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

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

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

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

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

Factor the expressions that are not already factored in \frac{x^{3}+3x^{2}+3x}{x\left(x+1\right)^{3}}.

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

Cancel out x in both numerator and denominator.

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

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

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