Find the Holes in the Graph f(x)=(-5x+10)/(2x^2-8)

$f (x) = \frac{- 5 x + 10}{2 x^{2} - 8}$

Step 1

Factor $5$ out of $- 5 x + 10$ .

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

$f (x) = \frac{5 (- x) + 10}{2 x^{2} - 8}$

Factor $5$ out of $10$ .

$f (x) = \frac{5 (- x) + 5 (2)}{2 x^{2} - 8}$

Factor $5$ out of $5 (- x) + 5 (2)$ .

$f (x) = \frac{5 (- x + 2)}{2 x^{2} - 8}$

Step 2

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

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

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

$f (x) = \frac{5 (- x + 2)}{2 (x^{2}) - 8}$

Factor $2$ out of $- 8$ .

$f (x) = \frac{5 (- x + 2)}{2 x^{2} + 2 \cdot - 4}$

Factor $2$ out of $2 x^{2} + 2 \cdot - 4$ .

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

Rewrite $4$ as $2^{2}$ .

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

Factor.

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

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

Remove unnecessary parentheses.

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

Step 3

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

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

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

Rewrite $2$ as $- 1 (- 2)$ .

$f (x) = \frac{5 (- (x) - 1 \cdot - 2)}{2 (x + 2) (x - 2)}$

Factor $- 1$ out of $- (x) - 1 (- 2)$ .

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

Rewrite $- (x - 2)$ as $- 1 (x - 2)$ .

$f (x) = \frac{5 (- 1 (x - 2))}{2 (x + 2) (x - 2)}$

Cancel the common factor.

$f (x) = \frac{5 (- 1 (x - 2))}{2 (x + 2) (x - 2)}$

Rewrite the expression.

$f (x) = \frac{5 \cdot (- 1)}{2 (x + 2)}$

Step 4

Simplify the numerator.

Multiply $5$ by $- 1$ .

$f (x) = \frac{5 (- 1)}{2 (x + 2)}$

Multiply $5$ by $- 1$ .

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

Step 5

Move the negative in front of the fraction.

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

Step 6

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

$x - 2$

Step 7

To find the coordinates of the holes, set each factor that was cancelled equal to $0$ , solve, and substitute back in to $- \frac{5}{2 (x + 2)}$ .

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

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

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

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

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

Simplify.

Add $2$ and $2$ .

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

Multiply $2$ by $4$ .

$- \frac{5}{8}$

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

$(2, - \frac{5}{8})$

Step 8

Find the Holes in the Graph f(x)=(-5x+10)/(2x^2-8)

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