Divide ((y^2-5y+4)/(y^2-1))÷((y^2-9)/(y^2+5y+4))

$\frac{y^{2} - 5 y + 4}{y^{2} - 1} \div \frac{y^{2} - 9}{y^{2} + 5 y + 4}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{y^{2} - 5 y + 4}{y^{2} - 1} \cdot \frac{y^{2} + 5 y + 4}{y^{2} - 9}$

Step 2

Factor $y^{2} - 5 y + 4$ 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 $4$ and whose sum is $- 5$ .

$- 4, - 1$

Write the factored form using these integers.

$\frac{(y - 4) (y - 1)}{y^{2} - 1} \cdot \frac{y^{2} + 5 y + 4}{y^{2} - 9}$

Step 3

Simplify the denominator.

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

$\frac{(y - 4) (y - 1)}{y^{2} - 1^{2}} \cdot \frac{y^{2} + 5 y + 4}{y^{2} - 9}$

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 $b = 1$ .

$\frac{(y - 4) (y - 1)}{(y + 1) (y - 1)} \cdot \frac{y^{2} + 5 y + 4}{y^{2} - 9}$

Step 4

Cancel the common factor of $y - 1$ .

Cancel the common factor.

$\frac{(y - 4) (y - 1)}{(y + 1) (y - 1)} \cdot \frac{y^{2} + 5 y + 4}{y^{2} - 9}$

Rewrite the expression.

$\frac{y - 4}{y + 1} \cdot \frac{y^{2} + 5 y + 4}{y^{2} - 9}$

Step 5

Factor $y^{2} + 5 y + 4$ 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 $4$ and whose sum is $5$ .

$1, 4$

Write the factored form using these integers.

$\frac{y - 4}{y + 1} \cdot \frac{(y + 1) (y + 4)}{y^{2} - 9}$

Step 6

Simplify the denominator.

Rewrite $9$ as $3^{2}$ .

$\frac{y - 4}{y + 1} \cdot \frac{(y + 1) (y + 4)}{y^{2} - 3^{2}}$

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 $b = 3$ .

$\frac{y - 4}{y + 1} \cdot \frac{(y + 1) (y + 4)}{(y + 3) (y - 3)}$

Step 7

Cancel the common factor of $y + 1$ .

Cancel the common factor.

$\frac{y - 4}{y + 1} \cdot \frac{(y + 1) (y + 4)}{(y + 3) (y - 3)}$

Rewrite the expression.

$(y - 4) \frac{y + 4}{(y + 3) (y - 3)}$

Step 8

Multiply $y - 4$ by $\frac{y + 4}{(y + 3) (y - 3)}$ .

$\frac{(y - 4) (y + 4)}{(y + 3) (y - 3)}$

Divide ((y^2-5y+4)/(y^2-1))÷((y^2-9)/(y^2+5y+4))

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