Simplify (2x^2-162)/(2x^2-8x-90)*(x-7)/(x^5+2x^4-63x^3)

$\frac{2 x^{2} - 162}{2 x^{2} - 8 x - 90} \cdot \frac{x - 7}{x^{5} + 2 x^{4} - 63 x^{3}}$

Step 1

Simplify the numerator.

Factor $2$ out of $2 x^{2} - 162$ .

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

$\frac{2 (x^{2}) - 162}{2 x^{2} - 8 x - 90} \cdot \frac{x - 7}{x^{5} + 2 x^{4} - 63 x^{3}}$

Factor $2$ out of $- 162$ .

$\frac{2 x^{2} + 2 \cdot - 81}{2 x^{2} - 8 x - 90} \cdot \frac{x - 7}{x^{5} + 2 x^{4} - 63 x^{3}}$

Factor $2$ out of $2 x^{2} + 2 \cdot - 81$ .

$\frac{2 (x^{2} - 81)}{2 x^{2} - 8 x - 90} \cdot \frac{x - 7}{x^{5} + 2 x^{4} - 63 x^{3}}$

Rewrite $81$ as $9^{2}$ .

$\frac{2 (x^{2} - 9^{2})}{2 x^{2} - 8 x - 90} \cdot \frac{x - 7}{x^{5} + 2 x^{4} - 63 x^{3}}$

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 $b = 9$ .

$\frac{2 (x + 9) (x - 9)}{2 x^{2} - 8 x - 90} \cdot \frac{x - 7}{x^{5} + 2 x^{4} - 63 x^{3}}$

Step 2

Simplify the denominator.

Factor $2$ out of $2 x^{2} - 8 x - 90$ .

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

$\frac{2 (x + 9) (x - 9)}{2 (x^{2}) - 8 x - 90} \cdot \frac{x - 7}{x^{5} + 2 x^{4} - 63 x^{3}}$

Factor $2$ out of $- 8 x$ .

$\frac{2 (x + 9) (x - 9)}{2 (x^{2}) + 2 (- 4 x) - 90} \cdot \frac{x - 7}{x^{5} + 2 x^{4} - 63 x^{3}}$

Factor $2$ out of $- 90$ .

$\frac{2 (x + 9) (x - 9)}{2 x^{2} + 2 (- 4 x) + 2 \cdot - 45} \cdot \frac{x - 7}{x^{5} + 2 x^{4} - 63 x^{3}}$

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

$\frac{2 (x + 9) (x - 9)}{2 (x^{2} - 4 x) + 2 \cdot - 45} \cdot \frac{x - 7}{x^{5} + 2 x^{4} - 63 x^{3}}$

Factor $2$ out of $2 (x^{2} - 4 x) + 2 \cdot - 45$ .

$\frac{2 (x + 9) (x - 9)}{2 (x^{2} - 4 x - 45)} \cdot \frac{x - 7}{x^{5} + 2 x^{4} - 63 x^{3}}$

Factor $x^{2} - 4 x - 45$ 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 $- 45$ and whose sum is $- 4$ .

$- 9, 5$

Write the factored form using these integers.

$\frac{2 (x + 9) (x - 9)}{2 ((x - 9) (x + 5))} \cdot \frac{x - 7}{x^{5} + 2 x^{4} - 63 x^{3}}$

$\frac{2 (x + 9) (x - 9)}{2 (x - 9) (x + 5)} \cdot \frac{x - 7}{x^{5} + 2 x^{4} - 63 x^{3}}$

Step 3

Simplify the denominator.

Factor $x^{3}$ out of $x^{5} + 2 x^{4} - 63 x^{3}$ .

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

$\frac{2 (x + 9) (x - 9)}{2 (x - 9) (x + 5)} \cdot \frac{x - 7}{x^{3} x^{2} + 2 x^{4} - 63 x^{3}}$

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

$\frac{2 (x + 9) (x - 9)}{2 (x - 9) (x + 5)} \cdot \frac{x - 7}{x^{3} x^{2} + x^{3} (2 x) - 63 x^{3}}$

Factor $x^{3}$ out of $- 63 x^{3}$ .

$\frac{2 (x + 9) (x - 9)}{2 (x - 9) (x + 5)} \cdot \frac{x - 7}{x^{3} x^{2} + x^{3} (2 x) + x^{3} \cdot - 63}$

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

$\frac{2 (x + 9) (x - 9)}{2 (x - 9) (x + 5)} \cdot \frac{x - 7}{x^{3} (x^{2} + 2 x) + x^{3} \cdot - 63}$

Factor $x^{3}$ out of $x^{3} (x^{2} + 2 x) + x^{3} \cdot - 63$ .

$\frac{2 (x + 9) (x - 9)}{2 (x - 9) (x + 5)} \cdot \frac{x - 7}{x^{3} (x^{2} + 2 x - 63)}$

Factor $x^{2} + 2 x - 63$ 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 $- 63$ and whose sum is $2$ .

$- 7, 9$

Write the factored form using these integers.

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

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

Step 4

Simplify terms.

Cancel the common factor of $x + 9$ .

Factor $x + 9$ out of $2 (x + 9) (x - 9)$ .

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

Factor $x + 9$ out of $x^{3} (x - 7) (x + 9)$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $\frac{2 (x - 9)}{2 (x - 9) (x + 5)}$ by $\frac{x - 7}{x^{3} (x - 7)}$ .

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x - 9$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{x - 7}{(x + 5) (x^{3} (x - 7))}$

Cancel the common factor of $x - 7$ .

Cancel the common factor.

$\frac{x - 7}{(x + 5) (x^{3} (x - 7))}$

Rewrite the expression.

$\frac{1}{(x + 5) (x^{3})}$

Reorder factors in $\frac{1}{(x + 5) (x^{3})}$ .

$\frac{1}{x^{3} (x + 5)}$

Simplify (2x^2-162)/(2x^2-8x-90)*(x-7)/(x^5+2x^4-63x^3)

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