Simplify (x^2-6x+5)/(x^2-9x+8)*(9x^2-576)/(x^3-5x^2)

$\frac{x^{2} - 6 x + 5}{x^{2} - 9 x + 8} \cdot \frac{9 x^{2} - 576}{x^{3} - 5 x^{2}}$

Factor $x^{2} - 6 x + 5$ 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 $5$ and whose sum is $- 6$ .

$- 5, - 1$

Write the factored form using these integers.

$\frac{(x - 5) (x - 1)}{x^{2} - 9 x + 8} \cdot \frac{9 x^{2} - 576}{x^{3} - 5 x^{2}}$

Factor $x^{2} - 9 x + 8$ 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 $8$ and whose sum is $- 9$ .

$- 8, - 1$

Write the factored form using these integers.

$\frac{(x - 5) (x - 1)}{(x - 8) (x - 1)} \cdot \frac{9 x^{2} - 576}{x^{3} - 5 x^{2}}$

Simplify the numerator.

Factor $9$ out of $9 x^{2} - 576$ .

Factor $9$ out of $9 x^{2}$ .

$\frac{(x - 5) (x - 1)}{(x - 8) (x - 1)} \cdot \frac{9 (x^{2}) - 576}{x^{3} - 5 x^{2}}$

Factor $9$ out of $- 576$ .

$\frac{(x - 5) (x - 1)}{(x - 8) (x - 1)} \cdot \frac{9 x^{2} + 9 \cdot - 64}{x^{3} - 5 x^{2}}$

Factor $9$ out of $9 x^{2} + 9 \cdot - 64$ .

$\frac{(x - 5) (x - 1)}{(x - 8) (x - 1)} \cdot \frac{9 (x^{2} - 64)}{x^{3} - 5 x^{2}}$

Rewrite $64$ as $8^{2}$ .

$\frac{(x - 5) (x - 1)}{(x - 8) (x - 1)} \cdot \frac{9 (x^{2} - 8^{2})}{x^{3} - 5 x^{2}}$

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 $b = 8$ .

$\frac{(x - 5) (x - 1)}{(x - 8) (x - 1)} \cdot \frac{9 (x + 8) (x - 8)}{x^{3} - 5 x^{2}}$

Simplify terms.

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

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

$\frac{(x - 5) (x - 1)}{(x - 8) (x - 1)} \cdot \frac{9 (x + 8) (x - 8)}{x^{2} x - 5 x^{2}}$

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

$\frac{(x - 5) (x - 1)}{(x - 8) (x - 1)} \cdot \frac{9 (x + 8) (x - 8)}{x^{2} x + x^{2} \cdot - 5}$

Factor $x^{2}$ out of $x^{2} x + x^{2} \cdot - 5$ .

$\frac{(x - 5) (x - 1)}{(x - 8) (x - 1)} \cdot \frac{9 (x + 8) (x - 8)}{x^{2} (x - 5)}$

Combine.

$\frac{(x - 5) (x - 1) (9 (x + 8) (x - 8))}{(x - 8) (x - 1) (x^{2} (x - 5))}$

Cancel the common factor of $x - 5$ .

Cancel the common factor.

$\frac{(x - 5) (x - 1) (9 (x + 8) (x - 8))}{(x - 8) (x - 1) (x^{2} (x - 5))}$

Rewrite the expression.

$\frac{(x - 1) (9 (x + 8) (x - 8))}{(x - 8) (x - 1) (x^{2})}$

Cancel the common factor of $x - 1$ .

Cancel the common factor.

$\frac{(x - 1) (9 (x + 8) (x - 8))}{(x - 8) (x - 1) x^{2}}$

Rewrite the expression.

$\frac{9 (x + 8) (x - 8)}{(x - 8) (x^{2})}$

Cancel the common factor of $x - 8$ .

Cancel the common factor.

$\frac{9 (x + 8) (x - 8)}{(x - 8) x^{2}}$

Rewrite the expression.

$\frac{9 (x + 8)}{x^{2}}$

Simplify (x^2-6x+5)/(x^2-9x+8)*(9x^2-576)/(x^3-5x^2)

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