Basic Math Examples

$\frac{p^{2} - 6 p + 9}{2 p^{2} - 7 p - 22} \cdot \frac{p^{2} - 6 p - 16}{3 p^{2} - 8 p - 3}$

Step 1

Factor using the perfect square rule.

Tap for more steps...

Step 1.1

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

$\frac{p^{2} - 6 p + 3^{2}}{2 p^{2} - 7 p - 22} \cdot \frac{p^{2} - 6 p - 16}{3 p^{2} - 8 p - 3}$

Step 1.2

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

$6 p = 2 \cdot p \cdot 3$

Step 1.3

Rewrite the polynomial.

$\frac{p^{2} - 2 \cdot p \cdot 3 + 3^{2}}{2 p^{2} - 7 p - 22} \cdot \frac{p^{2} - 6 p - 16}{3 p^{2} - 8 p - 3}$

Step 1.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = p$ and $b = 3$ .

$\frac{{(p - 3)}^{2}}{2 p^{2} - 7 p - 22} \cdot \frac{p^{2} - 6 p - 16}{3 p^{2} - 8 p - 3}$

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 $a \cdot c = 2 \cdot - 22 = - 44$ and whose sum is $b = - 7$ .

Tap for more steps...

Step 2.1.1

Factor $- 7$ out of $- 7 p$ .

$\frac{{(p - 3)}^{2}}{2 p^{2} - 7 (p) - 22} \cdot \frac{p^{2} - 6 p - 16}{3 p^{2} - 8 p - 3}$

Step 2.1.2

Rewrite $- 7$ as $4$ plus $- 11$

$\frac{{(p - 3)}^{2}}{2 p^{2} + (4 - 11) p - 22} \cdot \frac{p^{2} - 6 p - 16}{3 p^{2} - 8 p - 3}$

Step 2.1.3

Apply the distributive property.

$\frac{{(p - 3)}^{2}}{2 p^{2} + 4 p - 11 p - 22} \cdot \frac{p^{2} - 6 p - 16}{3 p^{2} - 8 p - 3}$

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.

$\frac{{(p - 3)}^{2}}{(2 p^{2} + 4 p) - 11 p - 22} \cdot \frac{p^{2} - 6 p - 16}{3 p^{2} - 8 p - 3}$

Step 2.2.2

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

$\frac{{(p - 3)}^{2}}{2 p (p + 2) - 11 (p + 2)} \cdot \frac{p^{2} - 6 p - 16}{3 p^{2} - 8 p - 3}$

Step 2.3

Factor the polynomial by factoring out the greatest common factor, $p + 2$ .

$\frac{{(p - 3)}^{2}}{(p + 2) (2 p - 11)} \cdot \frac{p^{2} - 6 p - 16}{3 p^{2} - 8 p - 3}$

Step 3

Factor $p^{2} - 6 p - 16$ 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 $- 16$ and whose sum is $- 6$ .

$- 8, 2$

Step 3.2

Write the factored form using these integers.

$\frac{{(p - 3)}^{2}}{(p + 2) (2 p - 11)} \cdot \frac{(p - 8) (p + 2)}{3 p^{2} - 8 p - 3}$

Step 4

Factor by grouping.

Tap for more steps...

Step 4.1

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 $a \cdot c = 3 \cdot - 3 = - 9$ and whose sum is $b = - 8$ .

Tap for more steps...

Step 4.1.1

Factor $- 8$ out of $- 8 p$ .

$\frac{{(p - 3)}^{2}}{(p + 2) (2 p - 11)} \cdot \frac{(p - 8) (p + 2)}{3 p^{2} - 8 (p) - 3}$

Step 4.1.2

Rewrite $- 8$ as $1$ plus $- 9$

$\frac{{(p - 3)}^{2}}{(p + 2) (2 p - 11)} \cdot \frac{(p - 8) (p + 2)}{3 p^{2} + (1 - 9) p - 3}$

Step 4.1.3

Apply the distributive property.

$\frac{{(p - 3)}^{2}}{(p + 2) (2 p - 11)} \cdot \frac{(p - 8) (p + 2)}{3 p^{2} + 1 p - 9 p - 3}$

Step 4.1.4

Multiply $p$ by $1$ .

$\frac{{(p - 3)}^{2}}{(p + 2) (2 p - 11)} \cdot \frac{(p - 8) (p + 2)}{3 p^{2} + p - 9 p - 3}$

Step 4.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 4.2.1

Group the first two terms and the last two terms.

$\frac{{(p - 3)}^{2}}{(p + 2) (2 p - 11)} \cdot \frac{(p - 8) (p + 2)}{(3 p^{2} + p) - 9 p - 3}$

Step 4.2.2

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

$\frac{{(p - 3)}^{2}}{(p + 2) (2 p - 11)} \cdot \frac{(p - 8) (p + 2)}{p (3 p + 1) - 3 (3 p + 1)}$

Step 4.3

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

$\frac{{(p - 3)}^{2}}{(p + 2) (2 p - 11)} \cdot \frac{(p - 8) (p + 2)}{(3 p + 1) (p - 3)}$

Step 5

Combine.

$\frac{{(p - 3)}^{2} ((p - 8) (p + 2))}{(p + 2) (2 p - 11) ((3 p + 1) (p - 3))}$

Step 6

Cancel the common factor of ${(p - 3)}^{2}$ and $p - 3$ .

Tap for more steps...

Step 6.1

Factor $p - 3$ out of ${(p - 3)}^{2} ((p - 8) (p + 2))$ .

$\frac{(p - 3) ((p - 3) ((p - 8) (p + 2)))}{(p + 2) (2 p - 11) ((3 p + 1) (p - 3))}$

Step 6.2

Cancel the common factors.

Tap for more steps...

Step 6.2.1

Factor $p - 3$ out of $(p + 2) (2 p - 11) ((3 p + 1) (p - 3))$ .

$\frac{(p - 3) ((p - 3) ((p - 8) (p + 2)))}{(p - 3) ((p + 2) (2 p - 11) (3 p + 1))}$

Step 6.2.2

Cancel the common factor.

$\frac{(p - 3) ((p - 3) ((p - 8) (p + 2)))}{(p - 3) ((p + 2) (2 p - 11) (3 p + 1))}$

Step 6.2.3

Rewrite the expression.

$\frac{(p - 3) ((p - 8) (p + 2))}{(p + 2) (2 p - 11) (3 p + 1)}$

Step 7

Cancel the common factor of $p + 2$ .

Tap for more steps...

Step 7.1

Cancel the common factor.

$\frac{(p - 3) ((p - 8) (p + 2))}{(p + 2) (2 p - 11) (3 p + 1)}$

Step 7.2

Rewrite the expression.

$\frac{(p - 3) (p - 8)}{(2 p - 11) (3 p + 1)}$