Simplify (5x^4-55x^3+120x^2)/(x^3-4x^2-32x)*(x^2-16)/(10x^2-70x+120)

$\frac{5 x^{4} - 55 x^{3} + 120 x^{2}}{x^{3} - 4 x^{2} - 32 x} \cdot \frac{x^{2} - 16}{10 x^{2} - 70 x + 120}$

Step 1

Simplify the numerator.

Factor $5 x^{2}$ out of $5 x^{4} - 55 x^{3} + 120 x^{2}$ .

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

$\frac{5 x^{2} (x^{2}) - 55 x^{3} + 120 x^{2}}{x^{3} - 4 x^{2} - 32 x} \cdot \frac{x^{2} - 16}{10 x^{2} - 70 x + 120}$

Factor $5 x^{2}$ out of $- 55 x^{3}$ .

$\frac{5 x^{2} (x^{2}) + 5 x^{2} (- 11 x) + 120 x^{2}}{x^{3} - 4 x^{2} - 32 x} \cdot \frac{x^{2} - 16}{10 x^{2} - 70 x + 120}$

Factor $5 x^{2}$ out of $120 x^{2}$ .

$\frac{5 x^{2} (x^{2}) + 5 x^{2} (- 11 x) + 5 x^{2} (24)}{x^{3} - 4 x^{2} - 32 x} \cdot \frac{x^{2} - 16}{10 x^{2} - 70 x + 120}$

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

$\frac{5 x^{2} (x^{2} - 11 x) + 5 x^{2} (24)}{x^{3} - 4 x^{2} - 32 x} \cdot \frac{x^{2} - 16}{10 x^{2} - 70 x + 120}$

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

$\frac{5 x^{2} (x^{2} - 11 x + 24)}{x^{3} - 4 x^{2} - 32 x} \cdot \frac{x^{2} - 16}{10 x^{2} - 70 x + 120}$

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.

$\frac{5 x^{2} ((x - 8) (x - 3))}{x^{3} - 4 x^{2} - 32 x} \cdot \frac{x^{2} - 16}{10 x^{2} - 70 x + 120}$

$\frac{5 x^{2} (x - 8) (x - 3)}{x^{3} - 4 x^{2} - 32 x} \cdot \frac{x^{2} - 16}{10 x^{2} - 70 x + 120}$

Step 2

Simplify the denominator.

Factor $x$ out of $x^{3} - 4 x^{2} - 32 x$ .

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

$\frac{5 x^{2} (x - 8) (x - 3)}{x \cdot x^{2} - 4 x^{2} - 32 x} \cdot \frac{x^{2} - 16}{10 x^{2} - 70 x + 120}$

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

$\frac{5 x^{2} (x - 8) (x - 3)}{x \cdot x^{2} + x (- 4 x) - 32 x} \cdot \frac{x^{2} - 16}{10 x^{2} - 70 x + 120}$

Factor $x$ out of $- 32 x$ .

$\frac{5 x^{2} (x - 8) (x - 3)}{x \cdot x^{2} + x (- 4 x) + x \cdot - 32} \cdot \frac{x^{2} - 16}{10 x^{2} - 70 x + 120}$

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

$\frac{5 x^{2} (x - 8) (x - 3)}{x (x^{2} - 4 x) + x \cdot - 32} \cdot \frac{x^{2} - 16}{10 x^{2} - 70 x + 120}$

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

$\frac{5 x^{2} (x - 8) (x - 3)}{x (x^{2} - 4 x - 32)} \cdot \frac{x^{2} - 16}{10 x^{2} - 70 x + 120}$

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 $- 4$ .

$- 8, 4$

Write the factored form using these integers.

$\frac{5 x^{2} (x - 8) (x - 3)}{x ((x - 8) (x + 4))} \cdot \frac{x^{2} - 16}{10 x^{2} - 70 x + 120}$

$\frac{5 x^{2} (x - 8) (x - 3)}{x (x - 8) (x + 4)} \cdot \frac{x^{2} - 16}{10 x^{2} - 70 x + 120}$

Step 3

Simplify the numerator.

Rewrite $16$ as $4^{2}$ .

$\frac{5 x^{2} (x - 8) (x - 3)}{x (x - 8) (x + 4)} \cdot \frac{x^{2} - 4^{2}}{10 x^{2} - 70 x + 120}$

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 $b = 4$ .

$\frac{5 x^{2} (x - 8) (x - 3)}{x (x - 8) (x + 4)} \cdot \frac{(x + 4) (x - 4)}{10 x^{2} - 70 x + 120}$

Step 4

Simplify the denominator.

Factor $10$ out of $10 x^{2} - 70 x + 120$ .

Factor $10$ out of $10 x^{2}$ .

$\frac{5 x^{2} (x - 8) (x - 3)}{x (x - 8) (x + 4)} \cdot \frac{(x + 4) (x - 4)}{10 (x^{2}) - 70 x + 120}$

Factor $10$ out of $- 70 x$ .

$\frac{5 x^{2} (x - 8) (x - 3)}{x (x - 8) (x + 4)} \cdot \frac{(x + 4) (x - 4)}{10 (x^{2}) + 10 (- 7 x) + 120}$

Factor $10$ out of $120$ .

$\frac{5 x^{2} (x - 8) (x - 3)}{x (x - 8) (x + 4)} \cdot \frac{(x + 4) (x - 4)}{10 x^{2} + 10 (- 7 x) + 10 \cdot 12}$

Factor $10$ out of $10 x^{2} + 10 (- 7 x)$ .

$\frac{5 x^{2} (x - 8) (x - 3)}{x (x - 8) (x + 4)} \cdot \frac{(x + 4) (x - 4)}{10 (x^{2} - 7 x) + 10 \cdot 12}$

Factor $10$ out of $10 (x^{2} - 7 x) + 10 \cdot 12$ .

$\frac{5 x^{2} (x - 8) (x - 3)}{x (x - 8) (x + 4)} \cdot \frac{(x + 4) (x - 4)}{10 (x^{2} - 7 x + 12)}$

Factor $x^{2} - 7 x + 12$ 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 $12$ and whose sum is $- 7$ .

$- 4, - 3$

Write the factored form using these integers.

$\frac{5 x^{2} (x - 8) (x - 3)}{x (x - 8) (x + 4)} \cdot \frac{(x + 4) (x - 4)}{10 ((x - 4) (x - 3))}$

$\frac{5 x^{2} (x - 8) (x - 3)}{x (x - 8) (x + 4)} \cdot \frac{(x + 4) (x - 4)}{10 (x - 4) (x - 3)}$

Step 5

Simplify terms.

Combine.

$\frac{5 x^{2} (x - 8) (x - 3) ((x + 4) (x - 4))}{x (x - 8) (x + 4) (10 (x - 4) (x - 3))}$

Cancel the common factor of $5$ and $10$ .

Factor $5$ out of $5 x^{2} (x - 8) (x - 3) ((x + 4) (x - 4))$ .

$\frac{5 (((x^{2} (x - 8)) (x - 3)) ((x + 4) (x - 4)))}{x (x - 8) (x + 4) (10 (x - 4) (x - 3))}$

Cancel the common factors.

Factor $5$ out of $x (x - 8) (x + 4) (10 (x - 4) (x - 3))$ .

$\frac{5 (((x^{2} (x - 8)) (x - 3)) ((x + 4) (x - 4)))}{5 (x (x - 8) (x + 4) (2 (x - 4) (x - 3)))}$

Cancel the common factor.

$\frac{5 (((x^{2} (x - 8)) (x - 3)) ((x + 4) (x - 4)))}{5 (x (x - 8) (x + 4) (2 (x - 4) (x - 3)))}$

Rewrite the expression.

$\frac{((x^{2} (x - 8)) (x - 3)) ((x + 4) (x - 4))}{x (x - 8) (x + 4) (2 (x - 4) (x - 3))}$

Cancel the common factor of $x^{2}$ and $x$ .

Factor $x$ out of $((x^{2} (x - 8)) (x - 3)) ((x + 4) (x - 4))$ .

$\frac{x (((x (x - 8)) (x - 3)) ((x + 4) (x - 4)))}{x (x - 8) (x + 4) (2 (x - 4) (x - 3))}$

Cancel the common factors.

Factor $x$ out of $x (x - 8) (x + 4) (2 (x - 4) (x - 3))$ .

$\frac{x (((x (x - 8)) (x - 3)) ((x + 4) (x - 4)))}{x (((x - 8) (x + 4)) (2 (x - 4) (x - 3)))}$

Cancel the common factor.

$\frac{x (((x (x - 8)) (x - 3)) ((x + 4) (x - 4)))}{x (((x - 8) (x + 4)) (2 (x - 4) (x - 3)))}$

Rewrite the expression.

$\frac{((x (x - 8)) (x - 3)) ((x + 4) (x - 4))}{((x - 8) (x + 4)) (2 (x - 4) (x - 3))}$

Cancel the common factor of $x - 8$ .

Cancel the common factor.

$\frac{x (x - 8) (x - 3) ((x + 4) (x - 4))}{(x - 8) (x + 4) (2 (x - 4) (x - 3))}$

Rewrite the expression.

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

Cancel the common factor of $x - 3$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{(x) ((x + 4) (x - 4))}{(x + 4) (2 (x - 4))}$

Cancel the common factor of $x + 4$ .

Cancel the common factor.

$\frac{x ((x + 4) (x - 4))}{(x + 4) (2 (x - 4))}$

Rewrite the expression.

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

Cancel the common factor of $x - 4$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{x}{2}$

Simplify (5x^4-55x^3+120x^2)/(x^3-4x^2-32x)*(x^2-16)/(10x^2-70x+120)

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