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

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

Factor $x^{2} + x - 2$ 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 $- 2$ and whose sum is $1$ .

$- 1, 2$

Write the factored form using these integers.

$f (x) = \frac{(x - 1) (x + 2)}{(x + 2) (x^{2} - 2 x - 15)}$

Factor.

Factor $x^{2} - 2 x - 15$ 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 $- 15$ and whose sum is $- 2$ .

$- 5, 3$

Write the factored form using these integers.

$f (x) = \frac{(x - 1) (x + 2)}{(x + 2) ((x - 5) (x + 3))}$

Remove unnecessary parentheses.

$f (x) = \frac{(x - 1) (x + 2)}{(x + 2) (x - 5) (x + 3)}$

Cancel the common factor of $x + 2$ .

Cancel the common factor.

$f (x) = \frac{(x - 1) (x + 2)}{(x + 2) (x - 5) (x + 3)}$

Rewrite the expression.

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

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

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

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

Substitute $- 2$ for $x$ in $\frac{x - 1}{(x - 5) (x + 3)}$ and simplify.

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

$\frac{- 2 - 1}{(- 2 - 5) (- 2 + 3)}$

Simplify.

Subtract $1$ from $- 2$ .

$\frac{- 3}{(- 2 - 5) (- 2 + 3)}$

Simplify the denominator.

Subtract $5$ from $- 2$ .

$\frac{- 3}{- 7 (- 2 + 3)}$

Add $- 2$ and $3$ .

$\frac{- 3}{- 7 \cdot 1}$

Multiply $- 7$ by $1$ .

$\frac{- 3}{- 7}$

Dividing two negative values results in a positive value.

$\frac{3}{7}$

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

$(- 2, \frac{3}{7})$

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

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