Find the Holes in the Graph f(x)=(2x-10)/(5x^2-51x+10)

$f (x) = \frac{2 x - 10}{5 x^{2} - 51 x + 10}$

Step 1

Factor $2$ out of $2 x - 10$ .

Factor $2$ out of $2 x$ .

$f (x) = \frac{2 (x) - 10}{5 x^{2} - 51 x + 10}$

Factor $2$ out of $- 10$ .

$f (x) = \frac{2 x + 2 \cdot - 5}{5 x^{2} - 51 x + 10}$

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

$f (x) = \frac{2 (x - 5)}{5 x^{2} - 51 x + 10}$

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 $a \cdot c = 5 \cdot 10 = 50$ and whose sum is $b = - 51$ .

Factor $- 51$ out of $- 51 x$ .

$f (x) = \frac{2 (x - 5)}{5 x^{2} - 51 x + 10}$

Rewrite $- 51$ as $- 1$ plus $- 50$

$f (x) = \frac{2 (x - 5)}{5 x^{2} + (- 1 - 50) x + 10}$

Apply the distributive property.

$f (x) = \frac{2 (x - 5)}{5 x^{2} - 1 x - 50 x + 10}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$f (x) = \frac{2 (x - 5)}{(5 x^{2} - 1 x) - 50 x + 10}$

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

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

Factor the polynomial by factoring out the greatest common factor, $5 x - 1$ .

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

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-10)/(5x^2-51x+10)

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