Simplify ((p^2-14p+40)/(16p^3))÷((3p^2-18p+24)/(2p^2-4p))

$\frac{p^{2} - 14 p + 40}{16 p^{3}} \div \frac{3 p^{2} - 18 p + 24}{2 p^{2} - 4 p}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{p^{2} - 14 p + 40}{16 p^{3}} \cdot \frac{2 p^{2} - 4 p}{3 p^{2} - 18 p + 24}$

Step 2

Factor $p^{2} - 14 p + 40$ using the AC method.

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 $40$ and whose sum is $- 14$ .

$- 10, - 4$

Write the factored form using these integers.

$\frac{(p - 10) (p - 4)}{16 p^{3}} \cdot \frac{2 p^{2} - 4 p}{3 p^{2} - 18 p + 24}$

Step 3

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

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

$\frac{(p - 10) (p - 4)}{16 p^{3}} \cdot \frac{2 p (p) - 4 p}{3 p^{2} - 18 p + 24}$

Factor $2 p$ out of $- 4 p$ .

$\frac{(p - 10) (p - 4)}{16 p^{3}} \cdot \frac{2 p (p) + 2 p (- 2)}{3 p^{2} - 18 p + 24}$

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

$\frac{(p - 10) (p - 4)}{16 p^{3}} \cdot \frac{2 p (p - 2)}{3 p^{2} - 18 p + 24}$

Step 4

Simplify the denominator.

Factor $3$ out of $3 p^{2} - 18 p + 24$ .

Factor $3$ out of $3 p^{2}$ .

$\frac{(p - 10) (p - 4)}{16 p^{3}} \cdot \frac{2 p (p - 2)}{3 (p^{2}) - 18 p + 24}$

Factor $3$ out of $- 18 p$ .

$\frac{(p - 10) (p - 4)}{16 p^{3}} \cdot \frac{2 p (p - 2)}{3 (p^{2}) + 3 (- 6 p) + 24}$

Factor $3$ out of $24$ .

$\frac{(p - 10) (p - 4)}{16 p^{3}} \cdot \frac{2 p (p - 2)}{3 p^{2} + 3 (- 6 p) + 3 \cdot 8}$

Factor $3$ out of $3 p^{2} + 3 (- 6 p)$ .

$\frac{(p - 10) (p - 4)}{16 p^{3}} \cdot \frac{2 p (p - 2)}{3 (p^{2} - 6 p) + 3 \cdot 8}$

Factor $3$ out of $3 (p^{2} - 6 p) + 3 \cdot 8$ .

$\frac{(p - 10) (p - 4)}{16 p^{3}} \cdot \frac{2 p (p - 2)}{3 (p^{2} - 6 p + 8)}$

Factor $p^{2} - 6 p + 8$ using the AC method.

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 $8$ and whose sum is $- 6$ .

$- 4, - 2$

Write the factored form using these integers.

$\frac{(p - 10) (p - 4)}{16 p^{3}} \cdot \frac{2 p (p - 2)}{3 ((p - 4) (p - 2))}$

$\frac{(p - 10) (p - 4)}{16 p^{3}} \cdot \frac{2 p (p - 2)}{3 (p - 4) (p - 2)}$

Step 5

Simplify terms.

Combine.

$\frac{(p - 10) (p - 4) (2 p (p - 2))}{16 p^{3} (3 (p - 4) (p - 2))}$

Cancel the common factor of $p - 4$ .

Cancel the common factor.

$\frac{(p - 10) (p - 4) (2 p (p - 2))}{16 p^{3} (3 (p - 4) (p - 2))}$

Rewrite the expression.

$\frac{(p - 10) (2 p (p - 2))}{16 p^{3} (3 (p - 2))}$

Cancel the common factor of $2$ and $16$ .

Factor $2$ out of $(p - 10) (2 p (p - 2))$ .

$\frac{2 ((p - 10) (p (p - 2)))}{16 p^{3} (3 (p - 2))}$

Cancel the common factors.

Factor $2$ out of $16 p^{3} (3 (p - 2))$ .

$\frac{2 ((p - 10) (p (p - 2)))}{2 (8 p^{3} (3 (p - 2)))}$

Cancel the common factor.

$\frac{2 ((p - 10) (p (p - 2)))}{2 (8 p^{3} (3 (p - 2)))}$

Rewrite the expression.

$\frac{(p - 10) (p (p - 2))}{8 p^{3} (3 (p - 2))}$

Cancel the common factor of $p$ and $p^{3}$ .

Factor $p$ out of $(p - 10) (p (p - 2))$ .

$\frac{p ((p - 10) (p - 2))}{8 p^{3} (3 (p - 2))}$

Cancel the common factors.

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

$\frac{p ((p - 10) (p - 2))}{p (8 p^{2} (3 (p - 2)))}$

Cancel the common factor.

$\frac{p ((p - 10) (p - 2))}{p (8 p^{2} (3 (p - 2)))}$

Rewrite the expression.

$\frac{(p - 10) (p - 2)}{8 p^{2} (3 (p - 2))}$

Cancel the common factor of $p - 2$ .

Cancel the common factor.

$\frac{(p - 10) (p - 2)}{8 p^{2} (3 (p - 2))}$

Rewrite the expression.

$\frac{p - 10}{8 p^{2} \cdot (3)}$

Multiply $3$ by $8$ .

$\frac{p - 10}{24 p^{2}}$

Simplify ((p^2-14p+40)/(16p^3))÷((3p^2-18p+24)/(2p^2-4p))

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