Simplify (x^5+9x^4-10x^3)/(9x(x-1)^2)*(6x^2-36x-240)/(x^2-100)

$\frac{x^{5} + 9 x^{4} - 10 x^{3}}{9 x {(x - 1)}^{2}} \cdot \frac{6 x^{2} - 36 x - 240}{x^{2} - 100}$

Step 1

Simplify the numerator.

Factor $x^{3}$ out of $x^{5} + 9 x^{4} - 10 x^{3}$ .

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

$\frac{x^{3} x^{2} + 9 x^{4} - 10 x^{3}}{9 x {(x - 1)}^{2}} \cdot \frac{6 x^{2} - 36 x - 240}{x^{2} - 100}$

Factor $x^{3}$ out of $9 x^{4}$ .

$\frac{x^{3} x^{2} + x^{3} (9 x) - 10 x^{3}}{9 x {(x - 1)}^{2}} \cdot \frac{6 x^{2} - 36 x - 240}{x^{2} - 100}$

Factor $x^{3}$ out of $- 10 x^{3}$ .

$\frac{x^{3} x^{2} + x^{3} (9 x) + x^{3} \cdot - 10}{9 x {(x - 1)}^{2}} \cdot \frac{6 x^{2} - 36 x - 240}{x^{2} - 100}$

Factor $x^{3}$ out of $x^{3} x^{2} + x^{3} (9 x)$ .

$\frac{x^{3} (x^{2} + 9 x) + x^{3} \cdot - 10}{9 x {(x - 1)}^{2}} \cdot \frac{6 x^{2} - 36 x - 240}{x^{2} - 100}$

Factor $x^{3}$ out of $x^{3} (x^{2} + 9 x) + x^{3} \cdot - 10$ .

$\frac{x^{3} (x^{2} + 9 x - 10)}{9 x {(x - 1)}^{2}} \cdot \frac{6 x^{2} - 36 x - 240}{x^{2} - 100}$

Factor $x^{2} + 9 x - 10$ 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 $- 10$ and whose sum is $9$ .

$- 1, 10$

Write the factored form using these integers.

$\frac{x^{3} ((x - 1) (x + 10))}{9 x {(x - 1)}^{2}} \cdot \frac{6 x^{2} - 36 x - 240}{x^{2} - 100}$

$\frac{x^{3} (x - 1) (x + 10)}{9 x {(x - 1)}^{2}} \cdot \frac{6 x^{2} - 36 x - 240}{x^{2} - 100}$

Step 2

Simplify the numerator.

Factor $6$ out of $6 x^{2} - 36 x - 240$ .

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

$\frac{x^{3} (x - 1) (x + 10)}{9 x {(x - 1)}^{2}} \cdot \frac{6 (x^{2}) - 36 x - 240}{x^{2} - 100}$

Factor $6$ out of $- 36 x$ .

$\frac{x^{3} (x - 1) (x + 10)}{9 x {(x - 1)}^{2}} \cdot \frac{6 (x^{2}) + 6 (- 6 x) - 240}{x^{2} - 100}$

Factor $6$ out of $- 240$ .

$\frac{x^{3} (x - 1) (x + 10)}{9 x {(x - 1)}^{2}} \cdot \frac{6 x^{2} + 6 (- 6 x) + 6 \cdot - 40}{x^{2} - 100}$

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

$\frac{x^{3} (x - 1) (x + 10)}{9 x {(x - 1)}^{2}} \cdot \frac{6 (x^{2} - 6 x) + 6 \cdot - 40}{x^{2} - 100}$

Factor $6$ out of $6 (x^{2} - 6 x) + 6 \cdot - 40$ .

$\frac{x^{3} (x - 1) (x + 10)}{9 x {(x - 1)}^{2}} \cdot \frac{6 (x^{2} - 6 x - 40)}{x^{2} - 100}$

Factor $x^{2} - 6 x - 40$ 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 $- 6$ .

$- 10, 4$

Write the factored form using these integers.

$\frac{x^{3} (x - 1) (x + 10)}{9 x {(x - 1)}^{2}} \cdot \frac{6 ((x - 10) (x + 4))}{x^{2} - 100}$

$\frac{x^{3} (x - 1) (x + 10)}{9 x {(x - 1)}^{2}} \cdot \frac{6 (x - 10) (x + 4)}{x^{2} - 100}$

Step 3

Simplify the denominator.

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

$\frac{x^{3} (x - 1) (x + 10)}{9 x {(x - 1)}^{2}} \cdot \frac{6 (x - 10) (x + 4)}{x^{2} - 10^{2}}$

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 $b = 10$ .

$\frac{x^{3} (x - 1) (x + 10)}{9 x {(x - 1)}^{2}} \cdot \frac{6 (x - 10) (x + 4)}{(x + 10) (x - 10)}$

Step 4

Simplify terms.

Combine.

$\frac{x^{3} (x - 1) (x + 10) (6 (x - 10) (x + 4))}{9 x {(x - 1)}^{2} ((x + 10) (x - 10))}$

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

Factor $x$ out of $x^{3} (x - 1) (x + 10) (6 (x - 10) (x + 4))$ .

$\frac{x (((x^{2} (x - 1)) (x + 10)) (6 (x - 10) (x + 4)))}{9 x {(x - 1)}^{2} ((x + 10) (x - 10))}$

Cancel the common factors.

Factor $x$ out of $9 x {(x - 1)}^{2} ((x + 10) (x - 10))$ .

$\frac{x (((x^{2} (x - 1)) (x + 10)) (6 (x - 10) (x + 4)))}{x (9 {(x - 1)}^{2} ((x + 10) (x - 10)))}$

Cancel the common factor.

$\frac{x (((x^{2} (x - 1)) (x + 10)) (6 (x - 10) (x + 4)))}{x (9 {(x - 1)}^{2} ((x + 10) (x - 10)))}$

Rewrite the expression.

$\frac{((x^{2} (x - 1)) (x + 10)) (6 (x - 10) (x + 4))}{9 {(x - 1)}^{2} ((x + 10) (x - 10))}$

Cancel the common factor of $x - 1$ and ${(x - 1)}^{2}$ .

Factor $x - 1$ out of $((x^{2} (x - 1)) (x + 10)) (6 (x - 10) (x + 4))$ .

$\frac{(x - 1) (((x^{2}) (x + 10)) (6 (x - 10) (x + 4)))}{9 {(x - 1)}^{2} ((x + 10) (x - 10))}$

Cancel the common factors.

Factor $x - 1$ out of $9 {(x - 1)}^{2} ((x + 10) (x - 10))$ .

$\frac{(x - 1) (((x^{2}) (x + 10)) (6 (x - 10) (x + 4)))}{(x - 1) (9 (x - 1) ((x + 10) (x - 10)))}$

Cancel the common factor.

$\frac{(x - 1) (((x^{2}) (x + 10)) (6 (x - 10) (x + 4)))}{(x - 1) (9 (x - 1) ((x + 10) (x - 10)))}$

Rewrite the expression.

$\frac{((x^{2}) (x + 10)) (6 (x - 10) (x + 4))}{9 (x - 1) ((x + 10) (x - 10))}$

Cancel the common factor of $x + 10$ .

Cancel the common factor.

$\frac{x^{2} (x + 10) (6 (x - 10) (x + 4))}{9 (x - 1) ((x + 10) (x - 10))}$

Rewrite the expression.

$\frac{(x^{2}) (6 (x - 10) (x + 4))}{9 (x - 1) (x - 10)}$

Cancel the common factor of $6$ and $9$ .

Factor $3$ out of $(x^{2}) (6 (x - 10) (x + 4))$ .

$\frac{3 (x^{2} (2 (x - 10) (x + 4)))}{9 (x - 1) (x - 10)}$

Cancel the common factors.

Factor $3$ out of $9 (x - 1) (x - 10)$ .

$\frac{3 (x^{2} (2 (x - 10) (x + 4)))}{3 (3 (x - 1) (x - 10))}$

Cancel the common factor.

$\frac{3 (x^{2} (2 (x - 10) (x + 4)))}{3 (3 (x - 1) (x - 10))}$

Rewrite the expression.

$\frac{x^{2} (2 (x - 10) (x + 4))}{3 (x - 1) (x - 10)}$

Cancel the common factor of $x - 10$ .

Cancel the common factor.

$\frac{x^{2} (2 (x - 10) (x + 4))}{3 (x - 1) (x - 10)}$

Rewrite the expression.

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

Step 5

Simplify the numerator.

Rewrite.

$\frac{x^{3} (x - 1) (x + 10) (6 (x - 10) (x + 4))}{3 (x - 1)}$

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

$\frac{x \cdot x^{2} (x - 1) (x + 10) (6 (x - 10) (x + 4))}{3 (x - 1)}$

Rewrite.

$\frac{x^{2} (x - 1) (x + 10) (6 (x - 10) (x + 4))}{3 (x - 1)}$

Multiply $(x^{2} (x - 1)) (x + 10)$ by $6 (x - 10) (x + 4)$ .

$\frac{((x^{2} (x - 1)) (x + 10)) (6 (x - 10) (x + 4))}{3 (x - 1)}$

Rewrite.

$\frac{x^{2} (x + 10) (6 (x - 10) (x + 4))}{3 (x - 1)}$

Multiply $(x^{2}) (x + 10)$ by $6 (x - 10) (x + 4)$ .

$\frac{((x^{2}) (x + 10)) (6 (x - 10) (x + 4))}{3 (x - 1)}$

Rewrite.

$\frac{x^{2} (6 (x - 10) (x + 4))}{3 (x - 1)}$

Multiply $x^{2}$ by $6 (x - 10) (x + 4)$ .

$\frac{(x^{2}) (6 (x - 10) (x + 4))}{3 (x - 1)}$

Factor $3$ out of $6$ .

$\frac{x^{2} (3 (2) (x - 10) (x + 4))}{3 (x - 1)}$

Rewrite.

$\frac{x^{2} (\frac{6}{3} (x - 10) (x + 4))}{3 (x - 1)}$

Divide $6$ by $3$ .

$\frac{x^{2} (2 (x - 10) (x + 4))}{3 (x - 1)}$

Rewrite.

$\frac{x^{2} (\frac{6}{3} (x + 4))}{3 (x - 1)}$

Divide $6$ by $3$ .

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

Remove unnecessary parentheses.

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

Step 6

Move $2$ to the left of $x^{2}$ .

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

Simplify (x^5+9x^4-10x^3)/(9x(x-1)^2)*(6x^2-36x-240)/(x^2-100)

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