Find the Holes in the Graph f(x)=(-5x-15)/(x^2+2x-3)

$f (x) = \frac{- 5 x - 15}{x^{2} + 2 x - 3}$

Step 1

Factor $5$ out of $- 5 x - 15$ .

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

$f (x) = \frac{5 (- x) - 15}{x^{2} + 2 x - 3}$

Factor $5$ out of $- 15$ .

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

Factor $5$ out of $5 (- x) + 5 (- 3)$ .

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

Step 2

Factor $x^{2} + 2 x - 3$ 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 $- 3$ and whose sum is $2$ .

$- 1, 3$

Write the factored form using these integers.

$f (x) = \frac{5 (- x - 3)}{(x - 1) (x + 3)}$

Step 3

Cancel the common factor of $- x - 3$ and $x + 3$ .

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

$f (x) = \frac{5 (- (x) - 3)}{(x - 1) (x + 3)}$

Rewrite $- 3$ as $- 1 (3)$ .

$f (x) = \frac{5 (- (x) - 1 \cdot 3)}{(x - 1) (x + 3)}$

Factor $- 1$ out of $- (x) - 1 (3)$ .

$f (x) = \frac{5 (- (x + 3))}{(x - 1) (x + 3)}$

Rewrite $- (x + 3)$ as $- 1 (x + 3)$ .

$f (x) = \frac{5 (- 1 (x + 3))}{(x - 1) (x + 3)}$

Cancel the common factor.

$f (x) = \frac{5 (- 1 (x + 3))}{(x - 1) (x + 3)}$

Rewrite the expression.

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

Step 4

Simplify the numerator.

Multiply $5$ by $- 1$ .

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

Multiply $5$ by $- 1$ .

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

Step 5

Move the negative in front of the fraction.

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

Step 6

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

$x + 3$

Step 7

To find the coordinates of the holes, set each factor that was cancelled equal to $0$ , solve, and substitute back in to $- \frac{5}{x - 1}$ .

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Subtract $3$ from both sides of the equation.

$x = - 3$

Substitute $- 3$ for $x$ in $- \frac{5}{x - 1}$ and simplify.

Substitute $- 3$ for $x$ to find the $y$ coordinate of the hole.

$- \frac{5}{- 3 - 1}$

Simplify.

Subtract $1$ from $- 3$ .

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

Move the negative in front of the fraction.

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

Multiply $- - \frac{5}{4}$ .

Multiply $- 1$ by $- 1$ .

$1 (\frac{5}{4})$

Multiply $\frac{5}{4}$ by $1$ .

$\frac{5}{4}$

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

$(- 3, \frac{5}{4})$

Step 8

Find the Holes in the Graph f(x)=(-5x-15)/(x^2+2x-3)

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