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

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

Step 1

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

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

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

Factor $x$ out of $- 2 x$ .

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

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

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

Step 2

Cancel the common factor of $x$ .

Cancel the common factor.

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

Rewrite the expression.

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

Step 3

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

$x$

Step 4

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

Set $x$ equal to $0$ .

$x = 0$

Substitute $0$ for $x$ in $\frac{x + 2}{x - 2}$ and simplify.

Substitute $0$ for $x$ to find the $y$ coordinate of the hole.

$\frac{0 + 2}{0 - 2}$

Simplify.

Cancel the common factor of $0 + 2$ and $0 - 2$ .

Factor $2$ out of $0$ .

$\frac{2 (0) + 2}{0 - 2}$

Factor $2$ out of $2$ .

$\frac{2 \cdot 0 + 2 \cdot 1}{0 - 2}$

Factor $2$ out of $2 \cdot 0 + 2 \cdot 1$ .

$\frac{2 \cdot (0 + 1)}{0 - 2}$

Cancel the common factors.

Factor $2$ out of $0$ .

$\frac{2 \cdot (0 + 1)}{2 (0) - 2}$

Factor $2$ out of $- 2$ .

$\frac{2 \cdot (0 + 1)}{2 \cdot 0 + 2 \cdot - 1}$

Factor $2$ out of $2 \cdot 0 + 2 \cdot - 1$ .

$\frac{2 \cdot (0 + 1)}{2 \cdot (0 - 1)}$

Cancel the common factor.

$\frac{2 \cdot (0 + 1)}{2 \cdot (0 - 1)}$

Rewrite the expression.

$\frac{0 + 1}{0 - 1}$

Cancel the common factor of $0 + 1$ and $0 - 1$ .

Rewrite $0$ as $- 1 (0)$ .

$\frac{- 1 (0) + 1}{0 - 1}$

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

$\frac{- 1 \cdot 0 - 1 \cdot - 1}{0 - 1}$

Factor $- 1$ out of $- 1 \cdot 0 - 1 \cdot - 1$ .

$\frac{- 1 \cdot (0 - 1)}{0 - 1}$

Cancel the common factor.

$\frac{- 1 \cdot (0 - 1)}{0 - 1}$

Divide $- 1$ by $1$ .

$- 1$

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

$(0, - 1)$

Step 5

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

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