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

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

Factor $y^{2} - y - 2$ 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.

$\frac{1}{y - 2} + \frac{2}{y + 1} - \frac{y + 2}{(y - 2) (y + 1)}$

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

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

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

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

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

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

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

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

$\frac{y + 1}{(y - 2) (y + 1)} + \frac{2 (y - 2)}{(y + 1) (y - 2)} - \frac{y + 2}{(y - 2) (y + 1)}$

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

$\frac{y + 1}{(y + 1) (y - 2)} + \frac{2 (y - 2)}{(y + 1) (y - 2)} - \frac{y + 2}{(y - 2) (y + 1)}$

Simplify terms.

Combine the numerators over the common denominator.

$\frac{y + 1 + 2 (y - 2)}{(y + 1) (y - 2)} - \frac{y + 2}{(y - 2) (y + 1)}$

Combine the numerators over the common denominator.

$\frac{y + 1 + 2 (y - 2) - (y + 2)}{(y + 1) (y - 2)}$

Simplify each term.

Apply the distributive property.

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

Multiply $2$ by $- 2$ .

$\frac{y + 1 + 2 y - 4 - (y + 2)}{(y + 1) (y - 2)}$

Apply the distributive property.

$\frac{y + 1 + 2 y - 4 - y - 1 \cdot 2}{(y + 1) (y - 2)}$

Multiply $- 1$ by $2$ .

$\frac{y + 1 + 2 y - 4 - y - 2}{(y + 1) (y - 2)}$

Simplify by adding terms.

Combine the opposite terms in $y + 1 + 2 y - 4 - y - 2$ .

Subtract $y$ from $y$ .

$\frac{1 + 2 y - 4 + 0 - 2}{(y + 1) (y - 2)}$

Add $1 + 2 y - 4$ and $0$ .

$\frac{1 + 2 y - 4 - 2}{(y + 1) (y - 2)}$

Simplify by subtracting numbers.

Subtract $4$ from $1$ .

$\frac{2 y - 3 - 2}{(y + 1) (y - 2)}$

Subtract $2$ from $- 3$ .

$\frac{2 y - 5}{(y + 1) (y - 2)}$

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

Related Questions

Support LCMGCF.com by disabling your adblocker.