Find the Holes in the Graph f(x)=(5x+25)/(x^2+x-20)

$f (x) = \frac{5 x + 25}{x^{2} + x - 20}$

Step 1

Factor $5$ out of $5 x + 25$ .

Factor $5$ out of $5 x$ .

$f (x) = \frac{5 (x) + 25}{x^{2} + x - 20}$

Factor $5$ out of $25$ .

$f (x) = \frac{5 x + 5 \cdot 5}{x^{2} + x - 20}$

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

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

Step 2

Factor $x^{2} + x - 20$ 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 $- 20$ and whose sum is $1$ .

$- 4, 5$

Write the factored form using these integers.

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

Step 3

Cancel the common factor of $x + 5$ .

Cancel the common factor.

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

Rewrite the expression.

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

Step 4

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

$x + 5$

Step 5

To find the coordinates of the holes, set each factor that was cancelled equal to $0$ , solve, and substitute back in to $\frac{5}{x - 4}$ .

Set $x + 5$ equal to $0$ .

$x + 5 = 0$

Subtract $5$ from both sides of the equation.

$x = - 5$

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

Substitute $- 5$ for $x$ to find the $y$ coordinate of the hole.

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

Simplify.

Subtract $4$ from $- 5$ .

$\frac{5}{- 9}$

Move the negative in front of the fraction.

$- \frac{5}{9}$

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

$(- 5, - \frac{5}{9})$

Step 6

Find the Holes in the Graph f(x)=(5x+25)/(x^2+x-20)

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