Finite Math Examples

$\frac{y^{2} + 14 y + 48}{y^{2} + 6 y - 7} \cdot \frac{y^{2} + 9 y - 10}{y^{2} + 12 y + 36}$

Step 1

Factor $y^{2} + 14 y + 48$ 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 $48$ and whose sum is $14$ .

$6, 8$

Step 1.2

Write the factored form using these integers.

$\frac{(y + 6) (y + 8)}{y^{2} + 6 y - 7} \cdot \frac{y^{2} + 9 y - 10}{y^{2} + 12 y + 36}$

Step 2

Factor $y^{2} + 6 y - 7$ using the AC method.

Tap for more steps...

Step 2.1

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 $- 7$ and whose sum is $6$ .

$- 1, 7$

Step 2.2

Write the factored form using these integers.

$\frac{(y + 6) (y + 8)}{(y - 1) (y + 7)} \cdot \frac{y^{2} + 9 y - 10}{y^{2} + 12 y + 36}$

Step 3

Factor $y^{2} + 9 y - 10$ using the AC method.

Tap for more steps...

Step 3.1

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 $- 10$ and whose sum is $9$ .

$- 1, 10$

Step 3.2

Write the factored form using these integers.

$\frac{(y + 6) (y + 8)}{(y - 1) (y + 7)} \cdot \frac{(y - 1) (y + 10)}{y^{2} + 12 y + 36}$

Step 4

Factor using the perfect square rule.

Tap for more steps...

Step 4.1

Rewrite $36$ as $6^{2}$ .

$\frac{(y + 6) (y + 8)}{(y - 1) (y + 7)} \cdot \frac{(y - 1) (y + 10)}{y^{2} + 12 y + 6^{2}}$

Step 4.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$12 y = 2 \cdot y \cdot 6$

Step 4.3

Rewrite the polynomial.

$\frac{(y + 6) (y + 8)}{(y - 1) (y + 7)} \cdot \frac{(y - 1) (y + 10)}{y^{2} + 2 \cdot y \cdot 6 + 6^{2}}$

Step 4.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = y$ and $b = 6$ .

$\frac{(y + 6) (y + 8)}{(y - 1) (y + 7)} \cdot \frac{(y - 1) (y + 10)}{{(y + 6)}^{2}}$

Step 5

Simplify terms.

Tap for more steps...

Step 5.1

Combine.

$\frac{(y + 6) (y + 8) ((y - 1) (y + 10))}{(y - 1) (y + 7) {(y + 6)}^{2}}$

Step 5.2

Cancel the common factor of $y + 6$ and ${(y + 6)}^{2}$ .

Tap for more steps...

Step 5.2.1

Factor $y + 6$ out of $(y + 6) (y + 8) ((y - 1) (y + 10))$ .

$\frac{(y + 6) ((y + 8) ((y - 1) (y + 10)))}{(y - 1) (y + 7) {(y + 6)}^{2}}$

Step 5.2.2

Cancel the common factors.

Tap for more steps...

Step 5.2.2.1

Factor $y + 6$ out of $(y - 1) (y + 7) {(y + 6)}^{2}$ .

$\frac{(y + 6) ((y + 8) ((y - 1) (y + 10)))}{(y + 6) ((y - 1) (y + 7) (y + 6))}$

Step 5.2.2.2

Cancel the common factor.

$\frac{(y + 6) ((y + 8) ((y - 1) (y + 10)))}{(y + 6) ((y - 1) (y + 7) (y + 6))}$

Step 5.2.2.3

Rewrite the expression.

$\frac{(y + 8) ((y - 1) (y + 10))}{(y - 1) (y + 7) (y + 6)}$

Step 5.3

Cancel the common factor of $y - 1$ .

Tap for more steps...

Step 5.3.1

Cancel the common factor.

$\frac{(y + 8) ((y - 1) (y + 10))}{(y - 1) (y + 7) (y + 6)}$

Step 5.3.2

Rewrite the expression.

$\frac{(y + 8) (y + 10)}{(y + 7) (y + 6)}$