Simplify ((6-x)/(64-x^2)*(x^3+12x^2+32x)/(6x^3-12x^2-144x))÷((x^2-15x+54)/(x^2-15x+54))

$\frac{6 - x}{64 - x^{2}} \cdot \frac{x^{3} + 12 x^{2} + 32 x}{6 x^{3} - 12 x^{2} - 144 x} \div \frac{x^{2} - 15 x + 54}{x^{2} - 15 x + 54}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{6 - x}{64 - x^{2}} \cdot \frac{x^{3} + 12 x^{2} + 32 x}{6 x^{3} - 12 x^{2} - 144 x} \frac{x^{2} - 15 x + 54}{x^{2} - 15 x + 54}$

Step 2

Reduce the expression by cancelling the common factors.

Cancel the common factor of $x^{2} - 15 x + 54$ .

Cancel the common factor.

$\frac{6 - x}{64 - x^{2}} \cdot \frac{x^{3} + 12 x^{2} + 32 x}{6 x^{3} - 12 x^{2} - 144 x} \frac{x^{2} - 15 x + 54}{x^{2} - 15 x + 54}$

Rewrite the expression.

$\frac{6 - x}{64 - x^{2}} \cdot \frac{x^{3} + 12 x^{2} + 32 x}{6 x^{3} - 12 x^{2} - 144 x} \cdot 1$

Multiply $\frac{6 - x}{64 - x^{2}} \cdot \frac{x^{3} + 12 x^{2} + 32 x}{6 x^{3} - 12 x^{2} - 144 x}$ by $1$ .

$\frac{6 - x}{64 - x^{2}} \cdot \frac{x^{3} + 12 x^{2} + 32 x}{6 x^{3} - 12 x^{2} - 144 x}$

Step 3

Simplify the denominator.

Rewrite $64$ as $8^{2}$ .

$\frac{6 - x}{8^{2} - x^{2}} \cdot \frac{x^{3} + 12 x^{2} + 32 x}{6 x^{3} - 12 x^{2} - 144 x}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 8$ and $b = x$ .

$\frac{6 - x}{(8 + x) (8 - x)} \cdot \frac{x^{3} + 12 x^{2} + 32 x}{6 x^{3} - 12 x^{2} - 144 x}$

Step 4

Simplify the numerator.

Factor $x$ out of $x^{3} + 12 x^{2} + 32 x$ .

Factor $x$ out of $x^{3}$ .

$\frac{6 - x}{(8 + x) (8 - x)} \cdot \frac{x \cdot x^{2} + 12 x^{2} + 32 x}{6 x^{3} - 12 x^{2} - 144 x}$

Factor $x$ out of $12 x^{2}$ .

$\frac{6 - x}{(8 + x) (8 - x)} \cdot \frac{x \cdot x^{2} + x (12 x) + 32 x}{6 x^{3} - 12 x^{2} - 144 x}$

Factor $x$ out of $32 x$ .

$\frac{6 - x}{(8 + x) (8 - x)} \cdot \frac{x \cdot x^{2} + x (12 x) + x \cdot 32}{6 x^{3} - 12 x^{2} - 144 x}$

Factor $x$ out of $x \cdot x^{2} + x (12 x)$ .

$\frac{6 - x}{(8 + x) (8 - x)} \cdot \frac{x (x^{2} + 12 x) + x \cdot 32}{6 x^{3} - 12 x^{2} - 144 x}$

Factor $x$ out of $x (x^{2} + 12 x) + x \cdot 32$ .

$\frac{6 - x}{(8 + x) (8 - x)} \cdot \frac{x (x^{2} + 12 x + 32)}{6 x^{3} - 12 x^{2} - 144 x}$

Factor $x^{2} + 12 x + 32$ 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 $32$ and whose sum is $12$ .

$4, 8$

Write the factored form using these integers.

$\frac{6 - x}{(8 + x) (8 - x)} \cdot \frac{x ((x + 4) (x + 8))}{6 x^{3} - 12 x^{2} - 144 x}$

$\frac{6 - x}{(8 + x) (8 - x)} \cdot \frac{x (x + 4) (x + 8)}{6 x^{3} - 12 x^{2} - 144 x}$

Step 5

Simplify the denominator.

Factor $6 x$ out of $6 x^{3} - 12 x^{2} - 144 x$ .

Factor $6 x$ out of $6 x^{3}$ .

$\frac{6 - x}{(8 + x) (8 - x)} \cdot \frac{x (x + 4) (x + 8)}{6 x (x^{2}) - 12 x^{2} - 144 x}$

Factor $6 x$ out of $- 12 x^{2}$ .

$\frac{6 - x}{(8 + x) (8 - x)} \cdot \frac{x (x + 4) (x + 8)}{6 x (x^{2}) + 6 x (- 2 x) - 144 x}$

Factor $6 x$ out of $- 144 x$ .

$\frac{6 - x}{(8 + x) (8 - x)} \cdot \frac{x (x + 4) (x + 8)}{6 x (x^{2}) + 6 x (- 2 x) + 6 x (- 24)}$

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

$\frac{6 - x}{(8 + x) (8 - x)} \cdot \frac{x (x + 4) (x + 8)}{6 x (x^{2} - 2 x) + 6 x (- 24)}$

Factor $6 x$ out of $6 x (x^{2} - 2 x) + 6 x (- 24)$ .

$\frac{6 - x}{(8 + x) (8 - x)} \cdot \frac{x (x + 4) (x + 8)}{6 x (x^{2} - 2 x - 24)}$

Factor $x^{2} - 2 x - 24$ 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 $- 24$ and whose sum is $- 2$ .

$- 6, 4$

Write the factored form using these integers.

$\frac{6 - x}{(8 + x) (8 - x)} \cdot \frac{x (x + 4) (x + 8)}{6 x ((x - 6) (x + 4))}$

$\frac{6 - x}{(8 + x) (8 - x)} \cdot \frac{x (x + 4) (x + 8)}{6 x (x - 6) (x + 4)}$

Step 6

Simplify terms.

Cancel the common factor of $x$ .

Cancel the common factor.

$\frac{6 - x}{(8 + x) (8 - x)} \cdot \frac{x (x + 4) (x + 8)}{6 x (x - 6) (x + 4)}$

Rewrite the expression.

$\frac{6 - x}{(8 + x) (8 - x)} \cdot \frac{(x + 4) (x + 8)}{(6 (x - 6)) (x + 4)}$

Cancel the common factor of $x + 4$ .

Cancel the common factor.

$\frac{6 - x}{(8 + x) (8 - x)} \cdot \frac{(x + 4) (x + 8)}{6 (x - 6) (x + 4)}$

Rewrite the expression.

$\frac{6 - x}{(8 + x) (8 - x)} \cdot \frac{x + 8}{6 (x - 6)}$

Multiply $\frac{6 - x}{(8 + x) (8 - x)}$ by $\frac{x + 8}{6 (x - 6)}$ .

$\frac{(6 - x) (x + 8)}{(8 + x) (8 - x) (6 (x - 6))}$

Cancel the common factor of $6 - x$ and $x - 6$ .

Rewrite $6$ as $- 1 (- 6)$ .

$\frac{(- 1 (- 6) - x) (x + 8)}{(8 + x) (8 - x) (6 (x - 6))}$

Factor $- 1$ out of $- x$ .

$\frac{(- 1 (- 6) - (x)) (x + 8)}{(8 + x) (8 - x) (6 (x - 6))}$

Factor $- 1$ out of $- 1 (- 6) - (x)$ .

$\frac{- 1 (- 6 + x) (x + 8)}{(8 + x) (8 - x) (6 (x - 6))}$

Reorder terms.

$\frac{- 1 (x - 6) (x + 8)}{(8 + x) (8 - x) (6 (x - 6))}$

Cancel the common factor.

$\frac{- 1 (x - 6) (x + 8)}{(8 + x) (8 - x) (6 (x - 6))}$

Rewrite the expression.

$\frac{- 1 (x + 8)}{(8 + x) (8 - x) \cdot (6)}$

Cancel the common factor of $x + 8$ and $8 + x$ .

Reorder terms.

$\frac{- 1 (x + 8)}{(x + 8) (8 - x) \cdot 6}$

Cancel the common factor.

$\frac{- 1 (x + 8)}{(x + 8) (8 - x) \cdot 6}$

Rewrite the expression.

$\frac{- 1}{(8 - x) \cdot 6}$

Simplify the expression.

Move $6$ to the left of $8 - x$ .

$\frac{- 1}{6 \cdot (8 - x)}$

Move the negative in front of the fraction.

$- \frac{1}{6 (8 - x)}$

Simplify ((6-x)/(64-x^2)*(x^3+12x^2+32x)/(6x^3-12x^2-144x))÷((x^2-15x+54)/(x^2-15x+54))

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