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

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

Step 1

Factor $x^{2} - x - 2$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 2$ and whose sum is $- 1$ .

$- 2, 1$

Write the factored form using these integers.

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

Step 2

To write $\frac{1}{x + 1}$ as a fraction with a common denominator, multiply by $\frac{x - 2}{x - 2}$ .

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

Step 3

To write $- \frac{x}{x - 2}$ as a fraction with a common denominator, multiply by $\frac{x + 1}{x + 1}$ .

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

Step 4

Write each expression with a common denominator of $(x + 1) (x - 2)$ , by multiplying each by an appropriate factor of $1$ .

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

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

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

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

Reorder the factors of $(x - 2) (x + 1)$ .

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

Step 5

Simplify terms.

Combine the numerators over the common denominator.

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

Combine the numerators over the common denominator.

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

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

Add $- 2$ and $2$ .

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

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

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

Step 6

Simplify each term.

Apply the distributive property.

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- 1$ by $1$ .

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

Step 7

Simplify by adding terms.

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

Subtract $x$ from $x$ .

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

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

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

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

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

Divide $0$ by $(x + 1) (x - 2)$ .

$0$

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

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