Find the Holes in the Graph f(x)=(2x^2+16x+30)/(5x^2-12x+4)

$f (x) = \frac{2 x^{2} + 16 x + 30}{5 x^{2} - 12 x + 4}$

Step 1

Factor $2 x^{2} + 16 x + 30$ .

Factor $2$ out of $2 x^{2} + 16 x + 30$ .

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

$f (x) = \frac{2 (x^{2}) + 16 x + 30}{5 x^{2} - 12 x + 4}$

Factor $2$ out of $16 x$ .

$f (x) = \frac{2 (x^{2}) + 2 (8 x) + 30}{5 x^{2} - 12 x + 4}$

Factor $2$ out of $30$ .

$f (x) = \frac{2 x^{2} + 2 (8 x) + 2 \cdot 15}{5 x^{2} - 12 x + 4}$

Factor $2$ out of $2 x^{2} + 2 (8 x)$ .

$f (x) = \frac{2 (x^{2} + 8 x) + 2 \cdot 15}{5 x^{2} - 12 x + 4}$

Factor $2$ out of $2 (x^{2} + 8 x) + 2 \cdot 15$ .

$f (x) = \frac{2 (x^{2} + 8 x + 15)}{5 x^{2} - 12 x + 4}$

Factor.

Factor $x^{2} + 8 x + 15$ 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 $8$ .

$3, 5$

Write the factored form using these integers.

$f (x) = \frac{2 ((x + 3) (x + 5))}{5 x^{2} - 12 x + 4}$

Remove unnecessary parentheses.

$f (x) = \frac{2 (x + 3) (x + 5)}{5 x^{2} - 12 x + 4}$

Step 2

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 $a \cdot c = 5 \cdot 4 = 20$ and whose sum is $b = - 12$ .

Factor $- 12$ out of $- 12 x$ .

$f (x) = \frac{2 (x + 3) (x + 5)}{5 x^{2} - 12 x + 4}$

Rewrite $- 12$ as $- 2$ plus $- 10$

$f (x) = \frac{2 (x + 3) (x + 5)}{5 x^{2} + (- 2 - 10) x + 4}$

Apply the distributive property.

$f (x) = \frac{2 (x + 3) (x + 5)}{5 x^{2} - 2 x - 10 x + 4}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$f (x) = \frac{2 (x + 3) (x + 5)}{(5 x^{2} - 2 x) - 10 x + 4}$

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

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

Factor the polynomial by factoring out the greatest common factor, $5 x - 2$ .

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

Step 3

Since no factors can be removed from the denominator, there are no holes in the graph.

No holes

Step 4

Find the Holes in the Graph f(x)=(2x^2+16x+30)/(5x^2-12x+4)

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