Find the Holes in the Graph f(x)=(-5x-10)/(x^2-4)

$f (x) = \frac{- 5 x - 10}{x^{2} - 4}$

Step 1

Factor $5$ out of $- 5 x - 10$ .

Factor $5$ out of $- 5 x$ .

$f (x) = \frac{5 (- x) - 10}{x^{2} - 4}$

Factor $5$ out of $- 10$ .

$f (x) = \frac{5 (- x) + 5 (- 2)}{x^{2} - 4}$

Factor $5$ out of $5 (- x) + 5 (- 2)$ .

$f (x) = \frac{5 (- x - 2)}{x^{2} - 4}$

Step 2

Factor $x^{2} - 4$ .

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

$f (x) = \frac{5 (- x - 2)}{x^{2} - 2^{2}}$

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 $b = 2$ .

$f (x) = \frac{5 (- x - 2)}{(x + 2) (x - 2)}$

Step 3

Cancel the common factor of $- x - 2$ and $x + 2$ .

Factor $- 1$ out of $- x$ .

$f (x) = \frac{5 (- (x) - 2)}{(x + 2) (x - 2)}$

Rewrite $- 2$ as $- 1 (2)$ .

$f (x) = \frac{5 (- (x) - 1 \cdot 2)}{(x + 2) (x - 2)}$

Factor $- 1$ out of $- (x) - 1 (2)$ .

$f (x) = \frac{5 (- (x + 2))}{(x + 2) (x - 2)}$

Rewrite $- (x + 2)$ as $- 1 (x + 2)$ .

$f (x) = \frac{5 (- 1 (x + 2))}{(x + 2) (x - 2)}$

Cancel the common factor.

$f (x) = \frac{5 (- 1 (x + 2))}{(x + 2) (x - 2)}$

Rewrite the expression.

$f (x) = \frac{5 \cdot (- 1)}{x - 2}$

Step 4

Simplify the numerator.

Multiply $5$ by $- 1$ .

$f (x) = \frac{5 (- 1)}{x - 2}$

Multiply $5$ by $- 1$ .

$f (x) = \frac{- 5}{x - 2}$

Step 5

Move the negative in front of the fraction.

$f (x) = - \frac{5}{x - 2}$

Step 6

To find the holes in the graph, look at the denominator factors that were cancelled.

$x + 2$

Step 7

To find the coordinates of the holes, set each factor that was cancelled equal to $0$ , solve, and substitute back in to $- \frac{5}{x - 2}$ .

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

Substitute $- 2$ for $x$ in $- \frac{5}{x - 2}$ and simplify.

Substitute $- 2$ for $x$ to find the $y$ coordinate of the hole.

$- \frac{5}{- 2 - 2}$

Simplify.

Subtract $2$ from $- 2$ .

$- \frac{5}{- 4}$

Move the negative in front of the fraction.

$- - \frac{5}{4}$

Multiply $- - \frac{5}{4}$ .

Multiply $- 1$ by $- 1$ .

$1 (\frac{5}{4})$

Multiply $\frac{5}{4}$ by $1$ .

$\frac{5}{4}$

The holes in the graph are the points where any of the cancelled factors are equal to $0$ .

$(- 2, \frac{5}{4})$

Step 8

Find the Holes in the Graph f(x)=(-5x-10)/(x^2-4)

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