Simplify (x^2-2x-15)/(x^3-11x^2+30x)

$\frac{x^{2} - 2 x - 15}{x^{3} - 11 x^{2} + 30 x}$

Step 1

Factor $x^{2} - 2 x - 15$ 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 $- 15$ and whose sum is $- 2$ .

$- 5, 3$

Write the factored form using these integers.

$\frac{(x - 5) (x + 3)}{x^{3} - 11 x^{2} + 30 x}$

Step 2

Simplify the denominator.

Factor $x$ out of $x^{3} - 11 x^{2} + 30 x$ .

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

$\frac{(x - 5) (x + 3)}{x \cdot x^{2} - 11 x^{2} + 30 x}$

Factor $x$ out of $- 11 x^{2}$ .

$\frac{(x - 5) (x + 3)}{x \cdot x^{2} + x (- 11 x) + 30 x}$

Factor $x$ out of $30 x$ .

$\frac{(x - 5) (x + 3)}{x \cdot x^{2} + x (- 11 x) + x \cdot 30}$

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

$\frac{(x - 5) (x + 3)}{x (x^{2} - 11 x) + x \cdot 30}$

Factor $x$ out of $x (x^{2} - 11 x) + x \cdot 30$ .

$\frac{(x - 5) (x + 3)}{x (x^{2} - 11 x + 30)}$

Factor $x^{2} - 11 x + 30$ 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 $- 11$ .

$- 6, - 5$

Write the factored form using these integers.

$\frac{(x - 5) (x + 3)}{x ((x - 6) (x - 5))}$

$\frac{(x - 5) (x + 3)}{x (x - 6) (x - 5)}$

Step 3

Cancel the common factor of $x - 5$ .

Cancel the common factor.

$\frac{(x - 5) (x + 3)}{x (x - 6) (x - 5)}$

Rewrite the expression.

$\frac{x + 3}{x (x - 6)}$

Simplify (x^2-2x-15)/(x^3-11x^2+30x)

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