Simplify (6x^2-12x-210)/(x^5-13x^4+42x^3)*(x-6)/(9x^2+9x-180)

$\frac{6 x^{2} - 12 x - 210}{x^{5} - 13 x^{4} + 42 x^{3}} \cdot \frac{x - 6}{9 x^{2} + 9 x - 180}$

Step 1

Simplify the numerator.

Factor $6$ out of $6 x^{2} - 12 x - 210$ .

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

$\frac{6 (x^{2}) - 12 x - 210}{x^{5} - 13 x^{4} + 42 x^{3}} \cdot \frac{x - 6}{9 x^{2} + 9 x - 180}$

Factor $6$ out of $- 12 x$ .

$\frac{6 (x^{2}) + 6 (- 2 x) - 210}{x^{5} - 13 x^{4} + 42 x^{3}} \cdot \frac{x - 6}{9 x^{2} + 9 x - 180}$

Factor $6$ out of $- 210$ .

$\frac{6 x^{2} + 6 (- 2 x) + 6 \cdot - 35}{x^{5} - 13 x^{4} + 42 x^{3}} \cdot \frac{x - 6}{9 x^{2} + 9 x - 180}$

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

$\frac{6 (x^{2} - 2 x) + 6 \cdot - 35}{x^{5} - 13 x^{4} + 42 x^{3}} \cdot \frac{x - 6}{9 x^{2} + 9 x - 180}$

Factor $6$ out of $6 (x^{2} - 2 x) + 6 \cdot - 35$ .

$\frac{6 (x^{2} - 2 x - 35)}{x^{5} - 13 x^{4} + 42 x^{3}} \cdot \frac{x - 6}{9 x^{2} + 9 x - 180}$

Factor $x^{2} - 2 x - 35$ 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 $- 35$ and whose sum is $- 2$ .

$- 7, 5$

Write the factored form using these integers.

$\frac{6 ((x - 7) (x + 5))}{x^{5} - 13 x^{4} + 42 x^{3}} \cdot \frac{x - 6}{9 x^{2} + 9 x - 180}$

$\frac{6 (x - 7) (x + 5)}{x^{5} - 13 x^{4} + 42 x^{3}} \cdot \frac{x - 6}{9 x^{2} + 9 x - 180}$

Step 2

Simplify the denominator.

Factor $x^{3}$ out of $x^{5} - 13 x^{4} + 42 x^{3}$ .

Factor $x^{3}$ out of $x^{5}$ .

$\frac{6 (x - 7) (x + 5)}{x^{3} x^{2} - 13 x^{4} + 42 x^{3}} \cdot \frac{x - 6}{9 x^{2} + 9 x - 180}$

Factor $x^{3}$ out of $- 13 x^{4}$ .

$\frac{6 (x - 7) (x + 5)}{x^{3} x^{2} + x^{3} (- 13 x) + 42 x^{3}} \cdot \frac{x - 6}{9 x^{2} + 9 x - 180}$

Factor $x^{3}$ out of $42 x^{3}$ .

$\frac{6 (x - 7) (x + 5)}{x^{3} x^{2} + x^{3} (- 13 x) + x^{3} \cdot 42} \cdot \frac{x - 6}{9 x^{2} + 9 x - 180}$

Factor $x^{3}$ out of $x^{3} x^{2} + x^{3} (- 13 x)$ .

$\frac{6 (x - 7) (x + 5)}{x^{3} (x^{2} - 13 x) + x^{3} \cdot 42} \cdot \frac{x - 6}{9 x^{2} + 9 x - 180}$

Factor $x^{3}$ out of $x^{3} (x^{2} - 13 x) + x^{3} \cdot 42$ .

$\frac{6 (x - 7) (x + 5)}{x^{3} (x^{2} - 13 x + 42)} \cdot \frac{x - 6}{9 x^{2} + 9 x - 180}$

Factor $x^{2} - 13 x + 42$ 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 $42$ and whose sum is $- 13$ .

$- 7, - 6$

Write the factored form using these integers.

$\frac{6 (x - 7) (x + 5)}{x^{3} ((x - 7) (x - 6))} \cdot \frac{x - 6}{9 x^{2} + 9 x - 180}$

$\frac{6 (x - 7) (x + 5)}{x^{3} (x - 7) (x - 6)} \cdot \frac{x - 6}{9 x^{2} + 9 x - 180}$

Step 3

Simplify the denominator.

Factor $9$ out of $9 x^{2} + 9 x - 180$ .

Factor $9$ out of $9 x^{2}$ .

$\frac{6 (x - 7) (x + 5)}{x^{3} (x - 7) (x - 6)} \cdot \frac{x - 6}{9 (x^{2}) + 9 x - 180}$

Factor $9$ out of $9 x$ .

$\frac{6 (x - 7) (x + 5)}{x^{3} (x - 7) (x - 6)} \cdot \frac{x - 6}{9 (x^{2}) + 9 (x) - 180}$

Factor $9$ out of $- 180$ .

$\frac{6 (x - 7) (x + 5)}{x^{3} (x - 7) (x - 6)} \cdot \frac{x - 6}{9 (x^{2}) + 9 x + 9 \cdot - 20}$

Factor $9$ out of $9 (x^{2}) + 9 x$ .

$\frac{6 (x - 7) (x + 5)}{x^{3} (x - 7) (x - 6)} \cdot \frac{x - 6}{9 (x^{2} + x) + 9 \cdot - 20}$

Factor $9$ out of $9 (x^{2} + x) + 9 \cdot - 20$ .

$\frac{6 (x - 7) (x + 5)}{x^{3} (x - 7) (x - 6)} \cdot \frac{x - 6}{9 (x^{2} + x - 20)}$

Factor $x^{2} + x - 20$ 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 $- 20$ and whose sum is $1$ .

$- 4, 5$

Write the factored form using these integers.

$\frac{6 (x - 7) (x + 5)}{x^{3} (x - 7) (x - 6)} \cdot \frac{x - 6}{9 ((x - 4) (x + 5))}$

$\frac{6 (x - 7) (x + 5)}{x^{3} (x - 7) (x - 6)} \cdot \frac{x - 6}{9 (x - 4) (x + 5)}$

Step 4

Simplify terms.

Cancel the common factor of $3 (x + 5)$ .

Factor $3 (x + 5)$ out of $6 (x - 7) (x + 5)$ .

$\frac{3 (x + 5) (2 (x - 7))}{x^{3} (x - 7) (x - 6)} \cdot \frac{x - 6}{9 (x - 4) (x + 5)}$

Factor $3 (x + 5)$ out of $9 (x - 4) (x + 5)$ .

$\frac{3 (x + 5) (2 (x - 7))}{x^{3} (x - 7) (x - 6)} \cdot \frac{x - 6}{3 (x + 5) (3 (x - 4))}$

Cancel the common factor.

$\frac{3 (x + 5) (2 (x - 7))}{x^{3} (x - 7) (x - 6)} \cdot \frac{x - 6}{3 (x + 5) (3 (x - 4))}$

Rewrite the expression.

$\frac{2 (x - 7)}{x^{3} (x - 7) (x - 6)} \cdot \frac{x - 6}{3 (x - 4)}$

Cancel the common factor of $x - 6$ .

Factor $x - 6$ out of $x^{3} (x - 7) (x - 6)$ .

$\frac{2 (x - 7)}{(x - 6) (x^{3} (x - 7))} \cdot \frac{x - 6}{3 (x - 4)}$

Cancel the common factor.

$\frac{2 (x - 7)}{(x - 6) (x^{3} (x - 7))} \cdot \frac{x - 6}{3 (x - 4)}$

Rewrite the expression.

$\frac{2 (x - 7)}{x^{3} (x - 7)} \cdot \frac{1}{3 (x - 4)}$

Multiply $\frac{2 (x - 7)}{x^{3} (x - 7)}$ by $\frac{1}{3 (x - 4)}$ .

$\frac{2 (x - 7)}{x^{3} (x - 7) (3 (x - 4))}$

Cancel the common factor of $x - 7$ .

Cancel the common factor.

$\frac{2 (x - 7)}{x^{3} (x - 7) (3 (x - 4))}$

Rewrite the expression.

$\frac{2}{x^{3} (3 (x - 4))}$

Step 5

Simplify the denominator.

Rewrite.

$\frac{2}{x^{3} (x - 7) (3 (x - 4))}$

Rewrite.

$\frac{2}{x^{3} (3 (x - 4))}$

Remove unnecessary parentheses.

$\frac{2}{x^{3} \cdot 3 (x - 4)}$

Step 6

Move $3$ to the left of $x^{3}$ .

$\frac{2}{3 x^{3} (x - 4)}$

Simplify (6x^2-12x-210)/(x^5-13x^4+42x^3)*(x-6)/(9x^2+9x-180)

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