Find the Holes in the Graph f(x)=(2x+8)/(x^2+6x+8)

$f (x) = \frac{2 x + 8}{x^{2} + 6 x + 8}$

Step 1

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

Factor $2$ out of $2 x$ .

$f (x) = \frac{2 (x) + 8}{x^{2} + 6 x + 8}$

Factor $2$ out of $8$ .

$f (x) = \frac{2 x + 2 \cdot 4}{x^{2} + 6 x + 8}$

Factor $2$ out of $2 x + 2 \cdot 4$ .

$f (x) = \frac{2 (x + 4)}{x^{2} + 6 x + 8}$

Step 2

Factor $x^{2} + 6 x + 8$ 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 $6$ .

$2, 4$

Write the factored form using these integers.

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

Step 3

Cancel the common factor of $x + 4$ .

Cancel the common factor.

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

Rewrite the expression.

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

Step 4

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

$x + 4$

Step 5

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

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Subtract $4$ from both sides of the equation.

$x = - 4$

Substitute $- 4$ for $x$ in $\frac{2}{x + 2}$ and simplify.

Substitute $- 4$ for $x$ to find the $y$ coordinate of the hole.

$\frac{2}{- 4 + 2}$

Simplify.

Cancel the common factor of $2$ and $- 4 + 2$ .

Factor $2$ out of $2$ .

$\frac{2 \cdot 1}{- 4 + 2}$

Cancel the common factors.

Factor $2$ out of $- 4$ .

$\frac{2 \cdot 1}{2 (- 2) + 2}$

Factor $2$ out of $2$ .

$\frac{2 \cdot 1}{2 \cdot - 2 + 2 \cdot 1}$

Factor $2$ out of $2 \cdot - 2 + 2 \cdot 1$ .

$\frac{2 \cdot 1}{2 \cdot (- 2 + 1)}$

Cancel the common factor.

$\frac{2 \cdot 1}{2 \cdot (- 2 + 1)}$

Rewrite the expression.

$\frac{1}{- 2 + 1}$

Add $- 2$ and $1$ .

$\frac{1}{- 1}$

Divide $1$ by $- 1$ .

$- 1$

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

$(- 4, - 1)$

Step 6

Find the Holes in the Graph f(x)=(2x+8)/(x^2+6x+8)

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