Algebra Examples

$\frac{p + 10}{9 - p} \cdot \frac{p^{2} - 5 p - 36}{4 p^{2} + 16 p}$

Step 1

Factor $p^{2} - 5 p - 36$ using the AC method.

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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 $- 36$ and whose sum is $- 5$ .

$- 9, 4$

Step 1.2

Write the factored form using these integers.

$\frac{p + 10}{9 - p} \cdot \frac{(p - 9) (p + 4)}{4 p^{2} + 16 p}$

Step 2

Simplify terms.

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Step 2.1

Factor $4 p$ out of $4 p^{2} + 16 p$ .

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Step 2.1.1

Factor $4 p$ out of $4 p^{2}$ .

$\frac{p + 10}{9 - p} \cdot \frac{(p - 9) (p + 4)}{4 p (p) + 16 p}$

Step 2.1.2

Factor $4 p$ out of $16 p$ .

$\frac{p + 10}{9 - p} \cdot \frac{(p - 9) (p + 4)}{4 p (p) + 4 p (4)}$

Step 2.1.3

Factor $4 p$ out of $4 p (p) + 4 p (4)$ .

$\frac{p + 10}{9 - p} \cdot \frac{(p - 9) (p + 4)}{4 p (p + 4)}$

Step 2.2

Cancel the common factor of $p + 4$ .

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Step 2.2.1

Cancel the common factor.

$\frac{p + 10}{9 - p} \cdot \frac{(p - 9) (p + 4)}{4 p (p + 4)}$

Step 2.2.2

Rewrite the expression.

$\frac{p + 10}{9 - p} \cdot \frac{p - 9}{4 p}$

Step 2.3

Multiply $\frac{p + 10}{9 - p}$ by $\frac{p - 9}{4 p}$ .

$\frac{(p + 10) (p - 9)}{(9 - p) (4 p)}$

Step 2.4

Cancel the common factor of $p - 9$ and $9 - p$ .

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Step 2.4.1

Factor $- 1$ out of $p$ .

$\frac{(p + 10) (- 1 (- p) - 9)}{(9 - p) (4 p)}$

Step 2.4.2

Rewrite $- 9$ as $- 1 (9)$ .

$\frac{(p + 10) (- 1 (- p) - 1 (9))}{(9 - p) (4 p)}$

Step 2.4.3

Factor $- 1$ out of $- 1 (- p) - 1 (9)$ .

$\frac{(p + 10) (- 1 (- p + 9))}{(9 - p) (4 p)}$

Step 2.4.4

Reorder terms.

$\frac{(p + 10) (- 1 (- p + 9))}{(- p + 9) \cdot 4 p}$

Step 2.4.5

Cancel the common factor.

$\frac{(p + 10) (- 1 (- p + 9))}{(- p + 9) \cdot 4 p}$

Step 2.4.6

Rewrite the expression.

$\frac{(p + 10) \cdot (- 1)}{4 p}$

Step 2.5

Simplify the expression.

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Step 2.5.1

Move $- 1$ to the left of $p + 10$ .

$\frac{- 1 \cdot (p + 10)}{4 p}$

Step 2.5.2

Move the negative in front of the fraction.

$- \frac{p + 10}{4 p}$