Find the Holes in the Graph f(x)=(3x^2-12x)/(5x-20)

$f (x) = \frac{3 x^{2} - 12 x}{5 x - 20}$

Step 1

Factor $3 x$ out of $3 x^{2} - 12 x$ .

Factor $3 x$ out of $3 x^{2}$ .

$f (x) = \frac{3 x (x) - 12 x}{5 x - 20}$

Factor $3 x$ out of $- 12 x$ .

$f (x) = \frac{3 x (x) + 3 x (- 4)}{5 x - 20}$

Factor $3 x$ out of $3 x (x) + 3 x (- 4)$ .

$f (x) = \frac{3 x (x - 4)}{5 x - 20}$

Step 2

Factor $5$ out of $5 x - 20$ .

Factor $5$ out of $5 x$ .

$f (x) = \frac{3 x (x - 4)}{5 (x) - 20}$

Factor $5$ out of $- 20$ .

$f (x) = \frac{3 x (x - 4)}{5 x + 5 \cdot - 4}$

Factor $5$ out of $5 x + 5 \cdot - 4$ .

$f (x) = \frac{3 x (x - 4)}{5 (x - 4)}$

Step 3

Cancel the common factor of $x - 4$ .

Cancel the common factor.

$f (x) = \frac{3 x (x - 4)}{5 (x - 4)}$

Rewrite the expression.

$f (x) = \frac{3 x}{5}$

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 $\frac{3 x}{5}$ .

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

$x - 4 = 0$

Add $4$ to both sides of the equation.

$x = 4$

Substitute $4$ for $x$ in $\frac{3 x}{5}$ and simplify.

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

$\frac{3 \cdot 4}{5}$

Multiply $3$ by $4$ .

$\frac{12}{5}$

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

$(4, \frac{12}{5})$

Step 6