Basic Math Examples

$\frac{3 p^{2} - 5 p - 2}{y^{2} + y - 2} \cdot \frac{y^{2} + 5 y - 6}{15 p^{2} + 8 p + 1} \div \frac{7 p^{2} - 13 p - 2}{10 p^{2} - 13 p - 3}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{3 p^{2} - 5 p - 2}{y^{2} + y - 2} \cdot \frac{y^{2} + 5 y - 6}{15 p^{2} + 8 p + 1} \frac{10 p^{2} - 13 p - 3}{7 p^{2} - 13 p - 2}$

Step 2

Factor by grouping.

Tap for more steps...

Step 2.1

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 $a \cdot c = 3 \cdot - 2 = - 6$ and whose sum is $b = - 5$ .

Tap for more steps...

Step 2.1.1

Factor $- 5$ out of $- 5 p$ .

$\frac{3 p^{2} - 5 (p) - 2}{y^{2} + y - 2} \cdot \frac{y^{2} + 5 y - 6}{15 p^{2} + 8 p + 1} \frac{10 p^{2} - 13 p - 3}{7 p^{2} - 13 p - 2}$

Step 2.1.2

Rewrite $- 5$ as $1$ plus $- 6$

$\frac{3 p^{2} + (1 - 6) p - 2}{y^{2} + y - 2} \cdot \frac{y^{2} + 5 y - 6}{15 p^{2} + 8 p + 1} \frac{10 p^{2} - 13 p - 3}{7 p^{2} - 13 p - 2}$

Step 2.1.3

Apply the distributive property.

$\frac{3 p^{2} + 1 p - 6 p - 2}{y^{2} + y - 2} \cdot \frac{y^{2} + 5 y - 6}{15 p^{2} + 8 p + 1} \frac{10 p^{2} - 13 p - 3}{7 p^{2} - 13 p - 2}$

Step 2.1.4

Multiply $p$ by $1$ .

$\frac{3 p^{2} + p - 6 p - 2}{y^{2} + y - 2} \cdot \frac{y^{2} + 5 y - 6}{15 p^{2} + 8 p + 1} \frac{10 p^{2} - 13 p - 3}{7 p^{2} - 13 p - 2}$

Step 2.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 2.2.1

Group the first two terms and the last two terms.

$\frac{(3 p^{2} + p) - 6 p - 2}{y^{2} + y - 2} \cdot \frac{y^{2} + 5 y - 6}{15 p^{2} + 8 p + 1} \frac{10 p^{2} - 13 p - 3}{7 p^{2} - 13 p - 2}$

Step 2.2.2

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

$\frac{p (3 p + 1) - 2 (3 p + 1)}{y^{2} + y - 2} \cdot \frac{y^{2} + 5 y - 6}{15 p^{2} + 8 p + 1} \frac{10 p^{2} - 13 p - 3}{7 p^{2} - 13 p - 2}$

Step 2.3

Factor the polynomial by factoring out the greatest common factor, $3 p + 1$ .

$\frac{(3 p + 1) (p - 2)}{y^{2} + y - 2} \cdot \frac{y^{2} + 5 y - 6}{15 p^{2} + 8 p + 1} \frac{10 p^{2} - 13 p - 3}{7 p^{2} - 13 p - 2}$

Step 3

Factor $y^{2} + y - 2$ 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 $- 2$ and whose sum is $1$ .

$- 1, 2$

Step 3.2

Write the factored form using these integers.

$\frac{(3 p + 1) (p - 2)}{(y - 1) (y + 2)} \cdot \frac{y^{2} + 5 y - 6}{15 p^{2} + 8 p + 1} \frac{10 p^{2} - 13 p - 3}{7 p^{2} - 13 p - 2}$

Step 4

Factor $y^{2} + 5 y - 6$ using the AC method.

Tap for more steps...

Step 4.1

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 $- 6$ and whose sum is $5$ .

$- 1, 6$

Step 4.2

Write the factored form using these integers.

$\frac{(3 p + 1) (p - 2)}{(y - 1) (y + 2)} \cdot \frac{(y - 1) (y + 6)}{15 p^{2} + 8 p + 1} \frac{10 p^{2} - 13 p - 3}{7 p^{2} - 13 p - 2}$

Step 5

Factor by grouping.

Tap for more steps...

Step 5.1

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 $a \cdot c = 15 \cdot 1 = 15$ and whose sum is $b = 8$ .

Tap for more steps...

Step 5.1.1

Factor $8$ out of $8 p$ .

$\frac{(3 p + 1) (p - 2)}{(y - 1) (y + 2)} \cdot \frac{(y - 1) (y + 6)}{15 p^{2} + 8 (p) + 1} \frac{10 p^{2} - 13 p - 3}{7 p^{2} - 13 p - 2}$

Step 5.1.2

Rewrite $8$ as $3$ plus $5$

$\frac{(3 p + 1) (p - 2)}{(y - 1) (y + 2)} \cdot \frac{(y - 1) (y + 6)}{15 p^{2} + (3 + 5) p + 1} \frac{10 p^{2} - 13 p - 3}{7 p^{2} - 13 p - 2}$

Step 5.1.3

Apply the distributive property.

$\frac{(3 p + 1) (p - 2)}{(y - 1) (y + 2)} \cdot \frac{(y - 1) (y + 6)}{15 p^{2} + 3 p + 5 p + 1} \frac{10 p^{2} - 13 p - 3}{7 p^{2} - 13 p - 2}$

Step 5.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 5.2.1

Group the first two terms and the last two terms.

$\frac{(3 p + 1) (p - 2)}{(y - 1) (y + 2)} \cdot \frac{(y - 1) (y + 6)}{(15 p^{2} + 3 p) + 5 p + 1} \frac{10 p^{2} - 13 p - 3}{7 p^{2} - 13 p - 2}$

Step 5.2.2

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

$\frac{(3 p + 1) (p - 2)}{(y - 1) (y + 2)} \cdot \frac{(y - 1) (y + 6)}{3 p (5 p + 1) + 1 (5 p + 1)} \frac{10 p^{2} - 13 p - 3}{7 p^{2} - 13 p - 2}$

Step 5.3

Factor the polynomial by factoring out the greatest common factor, $5 p + 1$ .

$\frac{(3 p + 1) (p - 2)}{(y - 1) (y + 2)} \cdot \frac{(y - 1) (y + 6)}{(5 p + 1) (3 p + 1)} \frac{10 p^{2} - 13 p - 3}{7 p^{2} - 13 p - 2}$

Step 6

Cancel the common factor of $3 p + 1$ .

Tap for more steps...

Step 6.1

Factor $3 p + 1$ out of $(5 p + 1) (3 p + 1)$ .

$\frac{(3 p + 1) (p - 2)}{(y - 1) (y + 2)} \cdot \frac{(y - 1) (y + 6)}{(3 p + 1) (5 p + 1)} \frac{10 p^{2} - 13 p - 3}{7 p^{2} - 13 p - 2}$

Step 6.2

Cancel the common factor.

$\frac{(3 p + 1) (p - 2)}{(y - 1) (y + 2)} \cdot \frac{(y - 1) (y + 6)}{(3 p + 1) (5 p + 1)} \frac{10 p^{2} - 13 p - 3}{7 p^{2} - 13 p - 2}$

Step 6.3

Rewrite the expression.

$\frac{p - 2}{(y - 1) (y + 2)} \cdot \frac{(y - 1) (y + 6)}{5 p + 1} \frac{10 p^{2} - 13 p - 3}{7 p^{2} - 13 p - 2}$

Step 7

Cancel the common factor of $y - 1$ .

Tap for more steps...

Step 7.1

Cancel the common factor.

$\frac{p - 2}{(y - 1) (y + 2)} \cdot \frac{(y - 1) (y + 6)}{5 p + 1} \frac{10 p^{2} - 13 p - 3}{7 p^{2} - 13 p - 2}$

Step 7.2

Rewrite the expression.

$\frac{p - 2}{y + 2} \cdot \frac{y + 6}{5 p + 1} \frac{10 p^{2} - 13 p - 3}{7 p^{2} - 13 p - 2}$

Step 8

Multiply $\frac{p - 2}{y + 2}$ by $\frac{y + 6}{5 p + 1}$ .

$\frac{(p - 2) (y + 6)}{(y + 2) (5 p + 1)} \cdot \frac{10 p^{2} - 13 p - 3}{7 p^{2} - 13 p - 2}$

Step 9

Factor by grouping.

Tap for more steps...

Step 9.1

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 $a \cdot c = 10 \cdot - 3 = - 30$ and whose sum is $b = - 13$ .

Tap for more steps...

Step 9.1.1

Factor $- 13$ out of $- 13 p$ .

$\frac{(p - 2) (y + 6)}{(y + 2) (5 p + 1)} \cdot \frac{10 p^{2} - 13 (p) - 3}{7 p^{2} - 13 p - 2}$

Step 9.1.2

Rewrite $- 13$ as $2$ plus $- 15$

$\frac{(p - 2) (y + 6)}{(y + 2) (5 p + 1)} \cdot \frac{10 p^{2} + (2 - 15) p - 3}{7 p^{2} - 13 p - 2}$

Step 9.1.3

Apply the distributive property.

$\frac{(p - 2) (y + 6)}{(y + 2) (5 p + 1)} \cdot \frac{10 p^{2} + 2 p - 15 p - 3}{7 p^{2} - 13 p - 2}$

Step 9.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 9.2.1

Group the first two terms and the last two terms.

$\frac{(p - 2) (y + 6)}{(y + 2) (5 p + 1)} \cdot \frac{(10 p^{2} + 2 p) - 15 p - 3}{7 p^{2} - 13 p - 2}$

Step 9.2.2

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

$\frac{(p - 2) (y + 6)}{(y + 2) (5 p + 1)} \cdot \frac{2 p (5 p + 1) - 3 (5 p + 1)}{7 p^{2} - 13 p - 2}$

Step 9.3

Factor the polynomial by factoring out the greatest common factor, $5 p + 1$ .

$\frac{(p - 2) (y + 6)}{(y + 2) (5 p + 1)} \cdot \frac{(5 p + 1) (2 p - 3)}{7 p^{2} - 13 p - 2}$

Step 10

Factor by grouping.

Tap for more steps...

Step 10.1

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 $a \cdot c = 7 \cdot - 2 = - 14$ and whose sum is $b = - 13$ .

Tap for more steps...

Step 10.1.1

Factor $- 13$ out of $- 13 p$ .

$\frac{(p - 2) (y + 6)}{(y + 2) (5 p + 1)} \cdot \frac{(5 p + 1) (2 p - 3)}{7 p^{2} - 13 (p) - 2}$

Step 10.1.2

Rewrite $- 13$ as $1$ plus $- 14$

$\frac{(p - 2) (y + 6)}{(y + 2) (5 p + 1)} \cdot \frac{(5 p + 1) (2 p - 3)}{7 p^{2} + (1 - 14) p - 2}$

Step 10.1.3

Apply the distributive property.

$\frac{(p - 2) (y + 6)}{(y + 2) (5 p + 1)} \cdot \frac{(5 p + 1) (2 p - 3)}{7 p^{2} + 1 p - 14 p - 2}$

Step 10.1.4

Multiply $p$ by $1$ .

$\frac{(p - 2) (y + 6)}{(y + 2) (5 p + 1)} \cdot \frac{(5 p + 1) (2 p - 3)}{7 p^{2} + p - 14 p - 2}$

Step 10.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 10.2.1

Group the first two terms and the last two terms.

$\frac{(p - 2) (y + 6)}{(y + 2) (5 p + 1)} \cdot \frac{(5 p + 1) (2 p - 3)}{(7 p^{2} + p) - 14 p - 2}$

Step 10.2.2

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

$\frac{(p - 2) (y + 6)}{(y + 2) (5 p + 1)} \cdot \frac{(5 p + 1) (2 p - 3)}{p (7 p + 1) - 2 (7 p + 1)}$

Step 10.3

Factor the polynomial by factoring out the greatest common factor, $7 p + 1$ .

$\frac{(p - 2) (y + 6)}{(y + 2) (5 p + 1)} \cdot \frac{(5 p + 1) (2 p - 3)}{(7 p + 1) (p - 2)}$

Step 11

Cancel the common factor of $p - 2$ .

Tap for more steps...

Step 11.1

Factor $p - 2$ out of $(7 p + 1) (p - 2)$ .

$\frac{(p - 2) (y + 6)}{(y + 2) (5 p + 1)} \cdot \frac{(5 p + 1) (2 p - 3)}{(p - 2) (7 p + 1)}$

Step 11.2

Cancel the common factor.

$\frac{(p - 2) (y + 6)}{(y + 2) (5 p + 1)} \cdot \frac{(5 p + 1) (2 p - 3)}{(p - 2) (7 p + 1)}$

Step 11.3

Rewrite the expression.

$\frac{y + 6}{(y + 2) (5 p + 1)} \cdot \frac{(5 p + 1) (2 p - 3)}{7 p + 1}$

Step 12

Cancel the common factor of $5 p + 1$ .

Tap for more steps...

Step 12.1

Factor $5 p + 1$ out of $(y + 2) (5 p + 1)$ .

$\frac{y + 6}{(5 p + 1) (y + 2)} \cdot \frac{(5 p + 1) (2 p - 3)}{7 p + 1}$

Step 12.2

Cancel the common factor.

$\frac{y + 6}{(5 p + 1) (y + 2)} \cdot \frac{(5 p + 1) (2 p - 3)}{7 p + 1}$

Step 12.3

Rewrite the expression.

$\frac{y + 6}{y + 2} \cdot \frac{2 p - 3}{7 p + 1}$

Step 13

Multiply $\frac{y + 6}{y + 2}$ by $\frac{2 p - 3}{7 p + 1}$ .

$\frac{(y + 6) (2 p - 3)}{(y + 2) (7 p + 1)}$