Find the Holes in the Graph (3x^2-9x+6)/(5x-10)

$\frac{3 x^{2} - 9 x + 6}{5 x - 10}$

Step 1

Factor $3 x^{2} - 9 x + 6$ .

Factor $3$ out of $3 x^{2} - 9 x + 6$ .

Factor $3$ out of $3 x^{2}$ .

$\frac{3 (x^{2}) - 9 x + 6}{5 x - 10}$

Factor $3$ out of $- 9 x$ .

$\frac{3 (x^{2}) + 3 (- 3 x) + 6}{5 x - 10}$

Factor $3$ out of $6$ .

$\frac{3 x^{2} + 3 (- 3 x) + 3 \cdot 2}{5 x - 10}$

Factor $3$ out of $3 x^{2} + 3 (- 3 x)$ .

$\frac{3 (x^{2} - 3 x) + 3 \cdot 2}{5 x - 10}$

Factor $3$ out of $3 (x^{2} - 3 x) + 3 \cdot 2$ .

$\frac{3 (x^{2} - 3 x + 2)}{5 x - 10}$

Factor.

Factor $x^{2} - 3 x + 2$ 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 $- 3$ .

$- 2, - 1$

Write the factored form using these integers.

$\frac{3 ((x - 2) (x - 1))}{5 x - 10}$

Remove unnecessary parentheses.

$\frac{3 (x - 2) (x - 1)}{5 x - 10}$

Step 2

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

Factor $5$ out of $5 x$ .

$\frac{3 (x - 2) (x - 1)}{5 (x) - 10}$

Factor $5$ out of $- 10$ .

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

Factor $5$ out of $5 x + 5 \cdot - 2$ .

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

Step 3

Cancel the common factor of $x - 2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Step 4

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

$x - 2$

Step 5

To find the coordinates of the holes, set each factor that was cancelled equal to $0$ , solve, and substitute back in to $\frac{3 (x - 1)}{5}$ .

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

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

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

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

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

Simplify.

Subtract $1$ from $2$ .

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

Multiply $3$ by $1$ .

$\frac{3}{5}$

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

$(2, \frac{3}{5})$

Step 6

Find the Holes in the Graph (3x^2-9x+6)/(5x-10)

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