Simplify (5x^3+20x^2-160x)/(x-6)*(x^2-36)/(15x^4+210x^3+720x^2)

$\frac{5 x^{3} + 20 x^{2} - 160 x}{x - 6} \cdot \frac{x^{2} - 36}{15 x^{4} + 210 x^{3} + 720 x^{2}}$

Step 1

Simplify the numerator.

Factor $5 x$ out of $5 x^{3} + 20 x^{2} - 160 x$ .

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

$\frac{5 x (x^{2}) + 20 x^{2} - 160 x}{x - 6} \cdot \frac{x^{2} - 36}{15 x^{4} + 210 x^{3} + 720 x^{2}}$

Factor $5 x$ out of $20 x^{2}$ .

$\frac{5 x (x^{2}) + 5 x (4 x) - 160 x}{x - 6} \cdot \frac{x^{2} - 36}{15 x^{4} + 210 x^{3} + 720 x^{2}}$

Factor $5 x$ out of $- 160 x$ .

$\frac{5 x (x^{2}) + 5 x (4 x) + 5 x (- 32)}{x - 6} \cdot \frac{x^{2} - 36}{15 x^{4} + 210 x^{3} + 720 x^{2}}$

Factor $5 x$ out of $5 x (x^{2}) + 5 x (4 x)$ .

$\frac{5 x (x^{2} + 4 x) + 5 x (- 32)}{x - 6} \cdot \frac{x^{2} - 36}{15 x^{4} + 210 x^{3} + 720 x^{2}}$

Factor $5 x$ out of $5 x (x^{2} + 4 x) + 5 x (- 32)$ .

$\frac{5 x (x^{2} + 4 x - 32)}{x - 6} \cdot \frac{x^{2} - 36}{15 x^{4} + 210 x^{3} + 720 x^{2}}$

Factor $x^{2} + 4 x - 32$ 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 $4$ .

$- 4, 8$

Write the factored form using these integers.

$\frac{5 x ((x - 4) (x + 8))}{x - 6} \cdot \frac{x^{2} - 36}{15 x^{4} + 210 x^{3} + 720 x^{2}}$

$\frac{5 x (x - 4) (x + 8)}{x - 6} \cdot \frac{x^{2} - 36}{15 x^{4} + 210 x^{3} + 720 x^{2}}$

Step 2

Simplify the numerator.

Rewrite $36$ as $6^{2}$ .

$\frac{5 x (x - 4) (x + 8)}{x - 6} \cdot \frac{x^{2} - 6^{2}}{15 x^{4} + 210 x^{3} + 720 x^{2}}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 6$ .

$\frac{5 x (x - 4) (x + 8)}{x - 6} \cdot \frac{(x + 6) (x - 6)}{15 x^{4} + 210 x^{3} + 720 x^{2}}$

Step 3

Simplify the denominator.

Factor $15 x^{2}$ out of $15 x^{4} + 210 x^{3} + 720 x^{2}$ .

Factor $15 x^{2}$ out of $15 x^{4}$ .

$\frac{5 x (x - 4) (x + 8)}{x - 6} \cdot \frac{(x + 6) (x - 6)}{15 x^{2} (x^{2}) + 210 x^{3} + 720 x^{2}}$

Factor $15 x^{2}$ out of $210 x^{3}$ .

$\frac{5 x (x - 4) (x + 8)}{x - 6} \cdot \frac{(x + 6) (x - 6)}{15 x^{2} (x^{2}) + 15 x^{2} (14 x) + 720 x^{2}}$

Factor $15 x^{2}$ out of $720 x^{2}$ .

$\frac{5 x (x - 4) (x + 8)}{x - 6} \cdot \frac{(x + 6) (x - 6)}{15 x^{2} (x^{2}) + 15 x^{2} (14 x) + 15 x^{2} (48)}$

Factor $15 x^{2}$ out of $15 x^{2} (x^{2}) + 15 x^{2} (14 x)$ .

$\frac{5 x (x - 4) (x + 8)}{x - 6} \cdot \frac{(x + 6) (x - 6)}{15 x^{2} (x^{2} + 14 x) + 15 x^{2} (48)}$

Factor $15 x^{2}$ out of $15 x^{2} (x^{2} + 14 x) + 15 x^{2} (48)$ .

$\frac{5 x (x - 4) (x + 8)}{x - 6} \cdot \frac{(x + 6) (x - 6)}{15 x^{2} (x^{2} + 14 x + 48)}$

Factor $x^{2} + 14 x + 48$ 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 $48$ and whose sum is $14$ .

$6, 8$

Write the factored form using these integers.

$\frac{5 x (x - 4) (x + 8)}{x - 6} \cdot \frac{(x + 6) (x - 6)}{15 x^{2} ((x + 6) (x + 8))}$

$\frac{5 x (x - 4) (x + 8)}{x - 6} \cdot \frac{(x + 6) (x - 6)}{15 x^{2} (x + 6) (x + 8)}$

Step 4

Simplify terms.

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

Factor $5 x (x + 8)$ out of $5 x (x - 4) (x + 8)$ .

$\frac{5 x (x + 8) (x - 4)}{x - 6} \cdot \frac{(x + 6) (x - 6)}{15 x^{2} (x + 6) (x + 8)}$

Factor $5 x (x + 8)$ out of $15 x^{2} (x + 6) (x + 8)$ .

$\frac{5 x (x + 8) (x - 4)}{x - 6} \cdot \frac{(x + 6) (x - 6)}{5 x (x + 8) (3 x (x + 6))}$

Cancel the common factor.

$\frac{5 x (x + 8) (x - 4)}{x - 6} \cdot \frac{(x + 6) (x - 6)}{5 x (x + 8) (3 x (x + 6))}$

Rewrite the expression.

$\frac{x - 4}{x - 6} \cdot \frac{(x + 6) (x - 6)}{3 x (x + 6)}$

Cancel the common factor of $x - 6$ .

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

$\frac{x - 4}{x - 6} \cdot \frac{(x - 6) (x + 6)}{3 x (x + 6)}$

Cancel the common factor.

$\frac{x - 4}{x - 6} \cdot \frac{(x - 6) (x + 6)}{3 x (x + 6)}$

Rewrite the expression.

$(x - 4) \cdot \frac{x + 6}{3 x (x + 6)}$

Cancel the common factor of $x + 6$ .

Cancel the common factor.

$(x - 4) \cdot \frac{x + 6}{3 x (x + 6)}$

Rewrite the expression.

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

Multiply $x - 4$ by $\frac{1}{3 x}$ .

$\frac{x - 4}{3 x}$

Simplify (5x^3+20x^2-160x)/(x-6)*(x^2-36)/(15x^4+210x^3+720x^2)

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