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

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

Step 1

Factor $2 x$ out of $2 x^{2} - 8 x$ .

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

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

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

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

Factor $2 x$ out of $2 x (x) + 2 x (- 4)$ .

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

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 $- 6$ .

$- 4, - 2$

Write the factored form using these integers.

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

Step 3

Cancel the common factor of $x - 4$ .

Cancel the common factor.

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

Rewrite the expression.

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

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 $\frac{2 x}{x - 2}$ .

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

$x - 4 = 0$

Add $4$ to both sides of the equation.

$x = 4$

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

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

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

Simplify.

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

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

$\frac{2 (4)}{4 - 2}$

Cancel the common factors.

Factor $2$ out of $4$ .

$\frac{2 (4)}{2 (2) - 2}$

Factor $2$ out of $- 2$ .

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

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

$\frac{2 (4)}{2 \cdot (2 - 1)}$

Cancel the common factor.

$\frac{2 \cdot 4}{2 \cdot (2 - 1)}$

Rewrite the expression.

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

Subtract $1$ from $2$ .

$\frac{4}{1}$

Divide $4$ by $1$ .

$4$

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

$(4, 4)$

Step 6

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

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