Simplify 1/x-2/(x+1)-x/((x+1)^2)

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

Step 1

Find the common denominator.

Multiply $\frac{1}{x}$ by $\frac{{(x + 1)}^{2}}{{(x + 1)}^{2}}$ .

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

Multiply $\frac{1}{x}$ by $\frac{{(x + 1)}^{2}}{{(x + 1)}^{2}}$ .

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

Multiply $\frac{2}{x + 1}$ by $\frac{x (x + 1)}{x (x + 1)}$ .

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

Multiply $\frac{2}{x + 1}$ by $\frac{x (x + 1)}{x (x + 1)}$ .

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

Multiply $\frac{x}{{(x + 1)}^{2}}$ by $\frac{x}{x}$ .

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

Multiply $\frac{x}{{(x + 1)}^{2}}$ by $\frac{x}{x}$ .

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

Reorder the factors of $(x + 1) (x (x + 1))$ .

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

Raise $x + 1$ to the power of $1$ .

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

Raise $x + 1$ to the power of $1$ .

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Add $1$ and $1$ .

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

Reorder the factors of ${(x + 1)}^{2} x$ .

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

Step 2

Combine the numerators over the common denominator.

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

Step 3

Simplify each term.

Rewrite ${(x + 1)}^{2}$ as $(x + 1) (x + 1)$ .

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

Expand $(x + 1) (x + 1)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Multiply $x$ by $1$ .

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

Multiply $x$ by $1$ .

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

Multiply $1$ by $1$ .

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

Add $x$ and $x$ .

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

Apply the distributive property.

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

Multiply $x$ by $x$ .

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

Multiply $x$ by $1$ .

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

Apply the distributive property.

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

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

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

Multiply $x$ by $x$ .

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

Step 4

Combine the opposite terms in $x^{2} + 2 x + 1 - 2 x^{2} - 2 x - x^{2}$ .

Subtract $2 x$ from $2 x$ .

$\frac{x^{2} + 1 - 2 x^{2} + 0 - x^{2}}{x {(x + 1)}^{2}}$

Add $x^{2} + 1 - 2 x^{2}$ and $0$ .

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

Subtract $x^{2}$ from $x^{2}$ .

$\frac{1 - 2 x^{2} + 0}{x {(x + 1)}^{2}}$

Add $1 - 2 x^{2}$ and $0$ .

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

Simplify 1/x-2/(x+1)-x/((x+1)^2)

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