Find the Holes in the Graph ((x-5)(x-2))/((x-2)(x+4))

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

Cancel the common factor of $x - 2$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

$x - 2$

To find the coordinates of the holes, set each factor that was cancelled equal to $0$ , solve, and substitute back in to $\frac{x - 5}{x + 4}$ .

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

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

Substitute $2$ for $x$ in $\frac{x - 5}{x + 4}$ and simplify.

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

$\frac{2 - 5}{2 + 4}$

Simplify.

Subtract $5$ from $2$ .

$\frac{- 3}{2 + 4}$

Add $2$ and $4$ .

$\frac{- 3}{6}$

Cancel the common factor of $- 3$ and $6$ .

Factor $3$ out of $- 3$ .

$\frac{3 (- 1)}{6}$

Cancel the common factors.

Factor $3$ out of $6$ .

$\frac{3 \cdot - 1}{3 \cdot 2}$

Cancel the common factor.

$\frac{3 \cdot - 1}{3 \cdot 2}$

Rewrite the expression.

$\frac{- 1}{2}$

Move the negative in front of the fraction.

$- \frac{1}{2}$

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

$(2, - \frac{1}{2})$