Find the Holes in the Graph f(x)=(-3x+3)/(x^2-6x+5)

$f (x) = \frac{- 3 x + 3}{x^{2} - 6 x + 5}$

Step 1

Factor $3$ out of $- 3 x + 3$ .

Factor $3$ out of $- 3 x$ .

$f (x) = \frac{3 (- x) + 3}{x^{2} - 6 x + 5}$

Factor $3$ out of $3$ .

$f (x) = \frac{3 (- x) + 3 (1)}{x^{2} - 6 x + 5}$

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

$f (x) = \frac{3 (- x + 1)}{x^{2} - 6 x + 5}$

Step 2

Factor $x^{2} - 6 x + 5$ 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 $5$ and whose sum is $- 6$ .

$- 5, - 1$

Write the factored form using these integers.

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

Step 3

Cancel the common factor of $- x + 1$ and $x - 1$ .

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

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

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

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

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

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 4

Simplify the numerator.

Multiply $3$ by $- 1$ .

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

Multiply $3$ by $- 1$ .

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

Step 5

Move the negative in front of the fraction.

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

Step 6

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

$x - 1$

Step 7

To find the coordinates of the holes, set each factor that was cancelled equal to $0$ , solve, and substitute back in to $- \frac{3}{x - 5}$ .

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

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

Substitute $1$ for $x$ to find the $y$ coordinate of the hole.

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

Simplify.

Subtract $5$ from $1$ .

$- \frac{3}{- 4}$

Move the negative in front of the fraction.

$- - \frac{3}{4}$

Multiply $- - \frac{3}{4}$ .

Multiply $- 1$ by $- 1$ .

$1 (\frac{3}{4})$

Multiply $\frac{3}{4}$ by $1$ .

$\frac{3}{4}$

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

$(1, \frac{3}{4})$

Step 8

Find the Holes in the Graph f(x)=(-3x+3)/(x^2-6x+5)

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