Find the Holes in the Graph f(x)=(5x+3)/(2x^2+30x+100)

$f (x) = \frac{5 x + 3}{2 x^{2} + 30 x + 100}$

Step 1

Factor $2 x^{2} + 30 x + 100$ .

Factor $2$ out of $2 x^{2} + 30 x + 100$ .

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

$f (x) = \frac{5 x + 3}{2 (x^{2}) + 30 x + 100}$

Factor $2$ out of $30 x$ .

$f (x) = \frac{5 x + 3}{2 (x^{2}) + 2 (15 x) + 100}$

Factor $2$ out of $100$ .

$f (x) = \frac{5 x + 3}{2 x^{2} + 2 (15 x) + 2 \cdot 50}$

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

$f (x) = \frac{5 x + 3}{2 (x^{2} + 15 x) + 2 \cdot 50}$

Factor $2$ out of $2 (x^{2} + 15 x) + 2 \cdot 50$ .

$f (x) = \frac{5 x + 3}{2 (x^{2} + 15 x + 50)}$

Factor.

Factor $x^{2} + 15 x + 50$ 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 $50$ and whose sum is $15$ .

$5, 10$

Write the factored form using these integers.

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

Remove unnecessary parentheses.

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

Step 2

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

No holes

Step 3

Find the Holes in the Graph f(x)=(5x+3)/(2x^2+30x+100)

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