Simplify 6/(x-8)(81-x^2)/(6x^3-24x^2-270x)(x^2-3x-40)/(x+9)

$\frac{6}{x - 8} \cdot \frac{81 - x^{2}}{6 x^{3} - 24 x^{2} - 270 x} \cdot \frac{x^{2} - 3 x - 40}{x + 9}$

Step 1

Factor $x^{2} - 3 x - 40$ 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 $- 3$ .

$- 8, 5$

Write the factored form using these integers.

$\frac{6}{x - 8} \cdot \frac{81 - x^{2}}{6 x^{3} - 24 x^{2} - 270 x} \cdot \frac{(x - 8) (x + 5)}{x + 9}$

Step 2

Simplify the numerator.

Rewrite $81$ as $9^{2}$ .

$\frac{6}{x - 8} \cdot \frac{9^{2} - x^{2}}{6 x^{3} - 24 x^{2} - 270 x} \cdot \frac{(x - 8) (x + 5)}{x + 9}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 9$ and $b = x$ .

$\frac{6}{x - 8} \cdot \frac{(9 + x) (9 - x)}{6 x^{3} - 24 x^{2} - 270 x} \cdot \frac{(x - 8) (x + 5)}{x + 9}$

Step 3

Simplify the denominator.

Factor $6 x$ out of $6 x^{3} - 24 x^{2} - 270 x$ .

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

$\frac{6}{x - 8} \cdot \frac{(9 + x) (9 - x)}{6 x (x^{2}) - 24 x^{2} - 270 x} \cdot \frac{(x - 8) (x + 5)}{x + 9}$

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

$\frac{6}{x - 8} \cdot \frac{(9 + x) (9 - x)}{6 x (x^{2}) + 6 x (- 4 x) - 270 x} \cdot \frac{(x - 8) (x + 5)}{x + 9}$

Factor $6 x$ out of $- 270 x$ .

$\frac{6}{x - 8} \cdot \frac{(9 + x) (9 - x)}{6 x (x^{2}) + 6 x (- 4 x) + 6 x (- 45)} \cdot \frac{(x - 8) (x + 5)}{x + 9}$

Factor $6 x$ out of $6 x (x^{2}) + 6 x (- 4 x)$ .

$\frac{6}{x - 8} \cdot \frac{(9 + x) (9 - x)}{6 x (x^{2} - 4 x) + 6 x (- 45)} \cdot \frac{(x - 8) (x + 5)}{x + 9}$

Factor $6 x$ out of $6 x (x^{2} - 4 x) + 6 x (- 45)$ .

$\frac{6}{x - 8} \cdot \frac{(9 + x) (9 - x)}{6 x (x^{2} - 4 x - 45)} \cdot \frac{(x - 8) (x + 5)}{x + 9}$

Factor $x^{2} - 4 x - 45$ 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 $- 45$ and whose sum is $- 4$ .

$- 9, 5$

Write the factored form using these integers.

$\frac{6}{x - 8} \cdot \frac{(9 + x) (9 - x)}{6 x ((x - 9) (x + 5))} \cdot \frac{(x - 8) (x + 5)}{x + 9}$

$\frac{6}{x - 8} \cdot \frac{(9 + x) (9 - x)}{6 x (x - 9) (x + 5)} \cdot \frac{(x - 8) (x + 5)}{x + 9}$

Step 4

Simplify terms.

Cancel the common factor of $6$ .

Factor $6$ out of $6 x (x - 9) (x + 5)$ .

$\frac{6}{x - 8} \cdot \frac{(9 + x) (9 - x)}{6 ((x (x - 9)) (x + 5))} \cdot \frac{(x - 8) (x + 5)}{x + 9}$

Cancel the common factor.

$\frac{6}{x - 8} \cdot \frac{(9 + x) (9 - x)}{6 ((x (x - 9)) (x + 5))} \cdot \frac{(x - 8) (x + 5)}{x + 9}$

Rewrite the expression.

$\frac{1}{x - 8} \cdot \frac{(9 + x) (9 - x)}{(x (x - 9)) (x + 5)} \cdot \frac{(x - 8) (x + 5)}{x + 9}$

Multiply $\frac{1}{x - 8}$ by $\frac{(9 + x) (9 - x)}{(x (x - 9)) (x + 5)}$ .

$\frac{(9 + x) (9 - x)}{(x - 8) ((x (x - 9)) (x + 5))} \cdot \frac{(x - 8) (x + 5)}{x + 9}$

Step 5

Remove parentheses.

$\frac{(9 + x) (9 - x)}{(x - 8) x (x - 9) (x + 5)} \cdot \frac{(x - 8) (x + 5)}{x + 9}$

Step 6

Simplify terms.

Cancel the common factor of $(x - 8) (x + 5)$ .

Factor $(x - 8) (x + 5)$ out of $(x - 8) x (x - 9) (x + 5)$ .

$\frac{(9 + x) (9 - x)}{(x - 8) (x + 5) (x (x - 9))} \cdot \frac{(x - 8) (x + 5)}{x + 9}$

Cancel the common factor.

$\frac{(9 + x) (9 - x)}{(x - 8) (x + 5) (x (x - 9))} \cdot \frac{(x - 8) (x + 5)}{x + 9}$

Rewrite the expression.

$\frac{(9 + x) (9 - x)}{x (x - 9)} \cdot \frac{1}{x + 9}$

Multiply $\frac{(9 + x) (9 - x)}{x (x - 9)}$ by $\frac{1}{x + 9}$ .

$\frac{(9 + x) (9 - x)}{x (x - 9) (x + 9)}$

Cancel the common factor of $9 + x$ and $x + 9$ .

Reorder terms.

$\frac{(x + 9) (9 - x)}{x (x - 9) (x + 9)}$

Cancel the common factor.

$\frac{(x + 9) (9 - x)}{x (x - 9) (x + 9)}$

Rewrite the expression.

$\frac{9 - x}{x (x - 9)}$

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

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

$\frac{- 1 (- 9) - x}{x (x - 9)}$

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

$\frac{- 1 (- 9) - (x)}{x (x - 9)}$

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

$\frac{- 1 (- 9 + x)}{x (x - 9)}$

Reorder terms.

$\frac{- 1 (x - 9)}{x (x - 9)}$

Cancel the common factor.

$\frac{- 1 (x - 9)}{x (x - 9)}$

Rewrite the expression.

$\frac{- 1}{x}$

Move the negative in front of the fraction.

$- \frac{1}{x}$

Simplify 6/(x-8)*(81-x^2)/(6x^3-24x^2-270x)*(x^2-3x-40)/(x+9)

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Simplify 6/(x-8)(81-x^2)/(6x^3-24x^2-270x)(x^2-3x-40)/(x+9)