Algebra Examples

$\frac{y^{2} - 2 y - 15}{6 y + 18} \cdot \frac{12 y + 36}{- y^{2} - 7 y + 60}$

Step 1

Factor $y^{2} - 2 y - 15$ using the AC method.

Tap for more steps...

Step 1.1

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 $- 15$ and whose sum is $- 2$ .

$- 5, 3$

Step 1.2

Write the factored form using these integers.

$\frac{(y - 5) (y + 3)}{6 y + 18} \cdot \frac{12 y + 36}{- y^{2} - 7 y + 60}$

Step 2

Simplify with factoring out.

Tap for more steps...

Step 2.1

Factor $6$ out of $6 y + 18$ .

Tap for more steps...

Step 2.1.1

Factor $6$ out of $6 y$ .

$\frac{(y - 5) (y + 3)}{6 (y) + 18} \cdot \frac{12 y + 36}{- y^{2} - 7 y + 60}$

Step 2.1.2

Factor $6$ out of $18$ .

$\frac{(y - 5) (y + 3)}{6 y + 6 \cdot 3} \cdot \frac{12 y + 36}{- y^{2} - 7 y + 60}$

Step 2.1.3

Factor $6$ out of $6 y + 6 \cdot 3$ .

$\frac{(y - 5) (y + 3)}{6 (y + 3)} \cdot \frac{12 y + 36}{- y^{2} - 7 y + 60}$

Step 2.2

Factor $12$ out of $12 y + 36$ .

Tap for more steps...

Step 2.2.1

Factor $12$ out of $12 y$ .

$\frac{(y - 5) (y + 3)}{6 (y + 3)} \cdot \frac{12 (y) + 36}{- y^{2} - 7 y + 60}$

Step 2.2.2

Factor $12$ out of $36$ .

$\frac{(y - 5) (y + 3)}{6 (y + 3)} \cdot \frac{12 y + 12 \cdot 3}{- y^{2} - 7 y + 60}$

Step 2.2.3

Factor $12$ out of $12 y + 12 \cdot 3$ .

$\frac{(y - 5) (y + 3)}{6 (y + 3)} \cdot \frac{12 (y + 3)}{- y^{2} - 7 y + 60}$

Step 3

Factor by grouping.

Tap for more steps...

Step 3.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot 60 = - 60$ and whose sum is $b = - 7$ .

Tap for more steps...

Step 3.1.1

Factor $- 7$ out of $- 7 y$ .

$\frac{(y - 5) (y + 3)}{6 (y + 3)} \cdot \frac{12 (y + 3)}{- y^{2} - 7 (y) + 60}$

Step 3.1.2

Rewrite $- 7$ as $5$ plus $- 12$

$\frac{(y - 5) (y + 3)}{6 (y + 3)} \cdot \frac{12 (y + 3)}{- y^{2} + (5 - 12) y + 60}$

Step 3.1.3

Apply the distributive property.

$\frac{(y - 5) (y + 3)}{6 (y + 3)} \cdot \frac{12 (y + 3)}{- y^{2} + 5 y - 12 y + 60}$

Step 3.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 3.2.1

Group the first two terms and the last two terms.

$\frac{(y - 5) (y + 3)}{6 (y + 3)} \cdot \frac{12 (y + 3)}{(- y^{2} + 5 y) - 12 y + 60}$

Step 3.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{(y - 5) (y + 3)}{6 (y + 3)} \cdot \frac{12 (y + 3)}{y (- y + 5) + 12 (- y + 5)}$

Step 3.3

Factor the polynomial by factoring out the greatest common factor, $- y + 5$ .

$\frac{(y - 5) (y + 3)}{6 (y + 3)} \cdot \frac{12 (y + 3)}{(- y + 5) (y + 12)}$

Step 4

Cancel the common factor of $6 (y + 3)$ .

Tap for more steps...

Step 4.1

Factor $6 (y + 3)$ out of $12 (y + 3)$ .

$\frac{(y - 5) (y + 3)}{6 (y + 3)} \cdot \frac{6 (y + 3) (2)}{(- y + 5) (y + 12)}$

Step 4.2

Cancel the common factor.

$\frac{(y - 5) (y + 3)}{6 (y + 3)} \cdot \frac{6 (y + 3) \cdot 2}{(- y + 5) (y + 12)}$

Step 4.3

Rewrite the expression.

$(y - 5) (y + 3) \cdot \frac{2}{(- y + 5) (y + 12)}$

Step 5

Expand $(y - 5) (y + 3)$ using the FOIL Method.

Tap for more steps...

Step 5.1

Apply the distributive property.

$(y (y + 3) - 5 (y + 3)) \cdot \frac{2}{(- y + 5) (y + 12)}$

Step 5.2

Apply the distributive property.

$(y \cdot y + y \cdot 3 - 5 (y + 3)) \cdot \frac{2}{(- y + 5) (y + 12)}$

Step 5.3

Apply the distributive property.

$(y \cdot y + y \cdot 3 - 5 y - 5 \cdot 3) \cdot \frac{2}{(- y + 5) (y + 12)}$

Step 6

Simplify and combine like terms.

Tap for more steps...

Step 6.1

Simplify each term.

Tap for more steps...

Step 6.1.1

Multiply $y$ by $y$ .

$(y^{2} + y \cdot 3 - 5 y - 5 \cdot 3) \cdot \frac{2}{(- y + 5) (y + 12)}$

Step 6.1.2

Move $3$ to the left of $y$ .

$(y^{2} + 3 \cdot y - 5 y - 5 \cdot 3) \cdot \frac{2}{(- y + 5) (y + 12)}$

Step 6.1.3

Multiply $- 5$ by $3$ .

$(y^{2} + 3 y - 5 y - 15) \cdot \frac{2}{(- y + 5) (y + 12)}$

Step 6.2

Subtract $5 y$ from $3 y$ .

$(y^{2} - 2 y - 15) \cdot \frac{2}{(- y + 5) (y + 12)}$

Step 7

Multiply $y^{2} - 2 y - 15$ by $\frac{2}{(- y + 5) (y + 12)}$ .

$\frac{(y^{2} - 2 y - 15) \cdot 2}{(- y + 5) (y + 12)}$

Step 8

Factor $y^{2} - 2 y - 15$ using the AC method.

Tap for more steps...

Step 8.1

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 $- 15$ and whose sum is $- 2$ .

$- 5, 3$

Step 8.2

Write the factored form using these integers.

$\frac{(y - 5) (y + 3) \cdot 2}{(- y + 5) (y + 12)}$

Step 9

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 9.1

Cancel the common factor of $y - 5$ and $- y + 5$ .

Tap for more steps...

Step 9.1.1

Factor $- 1$ out of $y$ .

$\frac{(- 1 (- y) - 5) (y + 3) \cdot 2}{(- y + 5) (y + 12)}$

Step 9.1.2

Rewrite $- 5$ as $- 1 (5)$ .

$\frac{(- 1 (- y) - 1 (5)) (y + 3) \cdot 2}{(- y + 5) (y + 12)}$

Step 9.1.3

Factor $- 1$ out of $- 1 (- y) - 1 (5)$ .

$\frac{- 1 (- y + 5) (y + 3) \cdot 2}{(- y + 5) (y + 12)}$

Step 9.1.4

Cancel the common factor.

$\frac{- 1 (- y + 5) (y + 3) \cdot 2}{(- y + 5) (y + 12)}$

Step 9.1.5

Rewrite the expression.

$\frac{(- 1 (y + 3)) \cdot 2}{y + 12}$

Step 9.2

Simplify the expression.

Tap for more steps...

Step 9.2.1

Multiply $2$ by $- 1$ .

$\frac{- 2 (y + 3)}{y + 12}$

Step 9.2.2

Move the negative in front of the fraction.

$- \frac{2 (y + 3)}{y + 12}$