Find the Holes in the Graph f(x)=(3x^2-22x+7)/(2x-14)

$f (x) = \frac{3 x^{2} - 22 x + 7}{2 x - 14}$

Step 1

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot 7 = 21$ and whose sum is $b = - 22$ .

Factor $- 22$ out of $- 22 x$ .

$f (x) = \frac{3 x^{2} - 22 x + 7}{2 x - 14}$

Rewrite $- 22$ as $- 1$ plus $- 21$

$f (x) = \frac{3 x^{2} + (- 1 - 21) x + 7}{2 x - 14}$

Apply the distributive property.

$f (x) = \frac{3 x^{2} - 1 x - 21 x + 7}{2 x - 14}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$f (x) = \frac{(3 x^{2} - 1 x) - 21 x + 7}{2 x - 14}$

Factor out the greatest common factor (GCF) from each group.

$f (x) = \frac{x (3 x - 1) - 7 (3 x - 1)}{2 x - 14}$

Factor the polynomial by factoring out the greatest common factor, $3 x - 1$ .

$f (x) = \frac{(3 x - 1) (x - 7)}{2 x - 14}$

Step 2

Factor $2$ out of $2 x - 14$ .

Factor $2$ out of $2 x$ .

$f (x) = \frac{(3 x - 1) (x - 7)}{2 (x) - 14}$

Factor $2$ out of $- 14$ .

$f (x) = \frac{(3 x - 1) (x - 7)}{2 x + 2 \cdot - 7}$

Factor $2$ out of $2 x + 2 \cdot - 7$ .

$f (x) = \frac{(3 x - 1) (x - 7)}{2 (x - 7)}$

Step 3

Cancel the common factor of $x - 7$ .

Cancel the common factor.

$f (x) = \frac{(3 x - 1) (x - 7)}{2 (x - 7)}$

Rewrite the expression.

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

Step 4

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

$x - 7$

Step 5

To find the coordinates of the holes, set each factor that was cancelled equal to $0$ , solve, and substitute back in to $\frac{3 x - 1}{2}$ .

Set $x - 7$ equal to $0$ .

$x - 7 = 0$

Add $7$ to both sides of the equation.

$x = 7$

Substitute $7$ for $x$ in $\frac{3 x - 1}{2}$ and simplify.

Substitute $7$ for $x$ to find the $y$ coordinate of the hole.

$\frac{3 \cdot 7 - 1}{2}$

Simplify.

Simplify the numerator.

Multiply $3$ by $7$ .

$\frac{21 - 1}{2}$

Subtract $1$ from $21$ .

$\frac{20}{2}$

Divide $20$ by $2$ .

$10$

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

$(7, 10)$

Step 6

Find the Holes in the Graph f(x)=(3x^2-22x+7)/(2x-14)

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