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

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

Step 1

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

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

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

Raise $x$ to the power of $1$ .

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

Factor $x$ out of $x^{1}$ .

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

Factor $x$ out of $x \cdot x + x \cdot 1$ .

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

Step 2

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

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

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

Factor $5$ out of $- 5$ .

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

Factor $5$ out of $5 (- x) + 5 (- 1)$ .

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

Step 3

Cancel the common factor of $x + 1$ and $- x - 1$ .

Factor $- 1$ out of $x$ .

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

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

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 4

Move $- 1$ to the left of $x$ .

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

Step 5

Move the negative in front of the fraction.

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

Step 6

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

$- x - 1$

Step 7

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

Set $- x - 1$ equal to $0$ .

$- x - 1 = 0$

Solve $- x - 1 = 0$ for $x$ .

Add $1$ to both sides of the equation.

$- x = 1$

Divide each term in $- x = 1$ by $- 1$ and simplify.

Divide each term in $- x = 1$ by $- 1$ .

$\frac{- x}{- 1} = \frac{1}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{1}{- 1}$

Divide $x$ by $1$ .

$x = \frac{1}{- 1}$

Simplify the right side.

Divide $1$ by $- 1$ .

$x = - 1$

Substitute $- 1$ for $x$ in $- \frac{x}{5}$ and simplify.

Substitute $- 1$ for $x$ to find the $y$ coordinate of the hole.

$- \frac{- 1}{5}$

Simplify.

Move the negative in front of the fraction.

$- - \frac{1}{5}$

Multiply $- - \frac{1}{5}$ .

Multiply $- 1$ by $- 1$ .

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

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

$\frac{1}{5}$

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

$(- 1, \frac{1}{5})$

Step 8

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

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