Simplify (y+3)/(y^2-2y-3)-7/(y^2-1)

$\frac{y + 3}{y^{2} - 2 y - 3} - \frac{7}{y^{2} - 1}$

Step 1

Simplify each term.

Factor $y^{2} - 2 y - 3$ 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 $- 3$ and whose sum is $- 2$ .

$- 3, 1$

Write the factored form using these integers.

$\frac{y + 3}{(y - 3) (y + 1)} - \frac{7}{y^{2} - 1}$

Simplify the denominator.

Rewrite $1$ as $1^{2}$ .

$\frac{y + 3}{(y - 3) (y + 1)} - \frac{7}{y^{2} - 1^{2}}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = y$ and $b = 1$ .

$\frac{y + 3}{(y - 3) (y + 1)} - \frac{7}{(y + 1) (y - 1)}$

Step 2

To write $\frac{y + 3}{(y - 3) (y + 1)}$ as a fraction with a common denominator, multiply by $\frac{y - 1}{y - 1}$ .

$\frac{y + 3}{(y - 3) (y + 1)} \cdot \frac{y - 1}{y - 1} - \frac{7}{(y + 1) (y - 1)}$

Step 3

To write $- \frac{7}{(y + 1) (y - 1)}$ as a fraction with a common denominator, multiply by $\frac{y - 3}{y - 3}$ .

$\frac{y + 3}{(y - 3) (y + 1)} \cdot \frac{y - 1}{y - 1} - \frac{7}{(y + 1) (y - 1)} \cdot \frac{y - 3}{y - 3}$

Step 4

Write each expression with a common denominator of $(y - 3) (y + 1) (y - 1)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{y + 3}{(y - 3) (y + 1)}$ by $\frac{y - 1}{y - 1}$ .

$\frac{(y + 3) (y - 1)}{(y - 3) (y + 1) (y - 1)} - \frac{7}{(y + 1) (y - 1)} \cdot \frac{y - 3}{y - 3}$

Multiply $\frac{7}{(y + 1) (y - 1)}$ by $\frac{y - 3}{y - 3}$ .

$\frac{(y + 3) (y - 1)}{(y - 3) (y + 1) (y - 1)} - \frac{7 (y - 3)}{(y + 1) (y - 1) (y - 3)}$

Reorder the factors of $(y - 3) (y + 1) (y - 1)$ .

$\frac{(y + 3) (y - 1)}{(y + 1) (y - 1) (y - 3)} - \frac{7 (y - 3)}{(y + 1) (y - 1) (y - 3)}$

Step 5

Combine the numerators over the common denominator.

$\frac{(y + 3) (y - 1) - 7 (y - 3)}{(y + 1) (y - 1) (y - 3)}$

Step 6

Simplify the numerator.

Expand $(y + 3) (y - 1)$ using the FOIL Method.

Apply the distributive property.

$\frac{y (y - 1) + 3 (y - 1) - 7 (y - 3)}{(y + 1) (y - 1) (y - 3)}$

Apply the distributive property.

$\frac{y \cdot y + y \cdot - 1 + 3 (y - 1) - 7 (y - 3)}{(y + 1) (y - 1) (y - 3)}$

Apply the distributive property.

$\frac{y \cdot y + y \cdot - 1 + 3 y + 3 \cdot - 1 - 7 (y - 3)}{(y + 1) (y - 1) (y - 3)}$

Simplify and combine like terms.

Simplify each term.

Multiply $y$ by $y$ .

$\frac{y^{2} + y \cdot - 1 + 3 y + 3 \cdot - 1 - 7 (y - 3)}{(y + 1) (y - 1) (y - 3)}$

Move $- 1$ to the left of $y$ .

$\frac{y^{2} - 1 \cdot y + 3 y + 3 \cdot - 1 - 7 (y - 3)}{(y + 1) (y - 1) (y - 3)}$

Rewrite $- 1 y$ as $- y$ .

$\frac{y^{2} - y + 3 y + 3 \cdot - 1 - 7 (y - 3)}{(y + 1) (y - 1) (y - 3)}$

Multiply $3$ by $- 1$ .

$\frac{y^{2} - y + 3 y - 3 - 7 (y - 3)}{(y + 1) (y - 1) (y - 3)}$

Add $- y$ and $3 y$ .

$\frac{y^{2} + 2 y - 3 - 7 (y - 3)}{(y + 1) (y - 1) (y - 3)}$

Apply the distributive property.

$\frac{y^{2} + 2 y - 3 - 7 y - 7 \cdot - 3}{(y + 1) (y - 1) (y - 3)}$

Multiply $- 7$ by $- 3$ .

$\frac{y^{2} + 2 y - 3 - 7 y + 21}{(y + 1) (y - 1) (y - 3)}$

Subtract $7 y$ from $2 y$ .

$\frac{y^{2} - 5 y - 3 + 21}{(y + 1) (y - 1) (y - 3)}$

Add $- 3$ and $21$ .

$\frac{y^{2} - 5 y + 18}{(y + 1) (y - 1) (y - 3)}$

Simplify (y+3)/(y^2-2y-3)-7/(y^2-1)

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