Simplify (5x^5-55x^4+120x^3)/(x^5+7x^4-30x^3)*(x^2-100)/(x^2-18x+80)

$\frac{5 x^{5} - 55 x^{4} + 120 x^{3}}{x^{5} + 7 x^{4} - 30 x^{3}} \cdot \frac{x^{2} - 100}{x^{2} - 18 x + 80}$

Simplify the numerator.

Factor $5 x^{3}$ out of $5 x^{5} - 55 x^{4} + 120 x^{3}$ .

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

$\frac{5 x^{3} (x^{2}) - 55 x^{4} + 120 x^{3}}{x^{5} + 7 x^{4} - 30 x^{3}} \cdot \frac{x^{2} - 100}{x^{2} - 18 x + 80}$

Factor $5 x^{3}$ out of $- 55 x^{4}$ .

$\frac{5 x^{3} (x^{2}) + 5 x^{3} (- 11 x) + 120 x^{3}}{x^{5} + 7 x^{4} - 30 x^{3}} \cdot \frac{x^{2} - 100}{x^{2} - 18 x + 80}$

Factor $5 x^{3}$ out of $120 x^{3}$ .

$\frac{5 x^{3} (x^{2}) + 5 x^{3} (- 11 x) + 5 x^{3} (24)}{x^{5} + 7 x^{4} - 30 x^{3}} \cdot \frac{x^{2} - 100}{x^{2} - 18 x + 80}$

Factor $5 x^{3}$ out of $5 x^{3} (x^{2}) + 5 x^{3} (- 11 x)$ .

$\frac{5 x^{3} (x^{2} - 11 x) + 5 x^{3} (24)}{x^{5} + 7 x^{4} - 30 x^{3}} \cdot \frac{x^{2} - 100}{x^{2} - 18 x + 80}$

Factor $5 x^{3}$ out of $5 x^{3} (x^{2} - 11 x) + 5 x^{3} (24)$ .

$\frac{5 x^{3} (x^{2} - 11 x + 24)}{x^{5} + 7 x^{4} - 30 x^{3}} \cdot \frac{x^{2} - 100}{x^{2} - 18 x + 80}$

Factor $x^{2} - 11 x + 24$ 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 $- 11$ .

$- 8, - 3$

Write the factored form using these integers.

$\frac{5 x^{3} ((x - 8) (x - 3))}{x^{5} + 7 x^{4} - 30 x^{3}} \cdot \frac{x^{2} - 100}{x^{2} - 18 x + 80}$

$\frac{5 x^{3} (x - 8) (x - 3)}{x^{5} + 7 x^{4} - 30 x^{3}} \cdot \frac{x^{2} - 100}{x^{2} - 18 x + 80}$

Simplify the denominator.

Factor $x^{3}$ out of $x^{5} + 7 x^{4} - 30 x^{3}$ .

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

$\frac{5 x^{3} (x - 8) (x - 3)}{x^{3} x^{2} + 7 x^{4} - 30 x^{3}} \cdot \frac{x^{2} - 100}{x^{2} - 18 x + 80}$

Factor $x^{3}$ out of $7 x^{4}$ .

$\frac{5 x^{3} (x - 8) (x - 3)}{x^{3} x^{2} + x^{3} (7 x) - 30 x^{3}} \cdot \frac{x^{2} - 100}{x^{2} - 18 x + 80}$

Factor $x^{3}$ out of $- 30 x^{3}$ .

$\frac{5 x^{3} (x - 8) (x - 3)}{x^{3} x^{2} + x^{3} (7 x) + x^{3} \cdot - 30} \cdot \frac{x^{2} - 100}{x^{2} - 18 x + 80}$

Factor $x^{3}$ out of $x^{3} x^{2} + x^{3} (7 x)$ .

$\frac{5 x^{3} (x - 8) (x - 3)}{x^{3} (x^{2} + 7 x) + x^{3} \cdot - 30} \cdot \frac{x^{2} - 100}{x^{2} - 18 x + 80}$

Factor $x^{3}$ out of $x^{3} (x^{2} + 7 x) + x^{3} \cdot - 30$ .

$\frac{5 x^{3} (x - 8) (x - 3)}{x^{3} (x^{2} + 7 x - 30)} \cdot \frac{x^{2} - 100}{x^{2} - 18 x + 80}$

Factor $x^{2} + 7 x - 30$ 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 $- 30$ and whose sum is $7$ .

$- 3, 10$

Write the factored form using these integers.

$\frac{5 x^{3} (x - 8) (x - 3)}{x^{3} ((x - 3) (x + 10))} \cdot \frac{x^{2} - 100}{x^{2} - 18 x + 80}$

$\frac{5 x^{3} (x - 8) (x - 3)}{x^{3} (x - 3) (x + 10)} \cdot \frac{x^{2} - 100}{x^{2} - 18 x + 80}$

Simplify the numerator.

Rewrite $100$ as $10^{2}$ .

$\frac{5 x^{3} (x - 8) (x - 3)}{x^{3} (x - 3) (x + 10)} \cdot \frac{x^{2} - 10^{2}}{x^{2} - 18 x + 80}$

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 $b = 10$ .

$\frac{5 x^{3} (x - 8) (x - 3)}{x^{3} (x - 3) (x + 10)} \cdot \frac{(x + 10) (x - 10)}{x^{2} - 18 x + 80}$

Factor $x^{2} - 18 x + 80$ 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 $80$ and whose sum is $- 18$ .

$- 10, - 8$

Write the factored form using these integers.

$\frac{5 x^{3} (x - 8) (x - 3)}{x^{3} (x - 3) (x + 10)} \cdot \frac{(x + 10) (x - 10)}{(x - 10) (x - 8)}$

Simplify terms.

Combine.

$\frac{5 x^{3} (x - 8) (x - 3) ((x + 10) (x - 10))}{x^{3} (x - 3) (x + 10) ((x - 10) (x - 8))}$

Cancel the common factor of $x^{3}$ .

Cancel the common factor.

$\frac{5 x^{3} (x - 8) (x - 3) ((x + 10) (x - 10))}{x^{3} (x - 3) (x + 10) ((x - 10) (x - 8))}$

Rewrite the expression.

$\frac{((5 (x - 8)) (x - 3)) ((x + 10) (x - 10))}{((x - 3) (x + 10)) ((x - 10) (x - 8))}$

Cancel the common factor of $x - 8$ .

Cancel the common factor.

$\frac{5 (x - 8) (x - 3) ((x + 10) (x - 10))}{(x - 3) (x + 10) ((x - 10) (x - 8))}$

Rewrite the expression.

$\frac{((5) (x - 3)) ((x + 10) (x - 10))}{((x - 3) (x + 10)) (x - 10)}$

Cancel the common factor of $x - 3$ .

Cancel the common factor.

$\frac{5 (x - 3) ((x + 10) (x - 10))}{(x - 3) (x + 10) (x - 10)}$

Rewrite the expression.

$\frac{(5) ((x + 10) (x - 10))}{(x + 10) (x - 10)}$

Cancel the common factor of $x + 10$ .

Cancel the common factor.

$\frac{5 ((x + 10) (x - 10))}{(x + 10) (x - 10)}$

Rewrite the expression.

$\frac{(5) (x - 10)}{x - 10}$

Cancel the common factor of $x - 10$ .

Cancel the common factor.

$\frac{5 (x - 10)}{x - 10}$

Divide $5$ by $1$ .

$5$

Simplify (5x^5-55x^4+120x^3)/(x^5+7x^4-30x^3)*(x^2-100)/(x^2-18x+80)

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