Algebra Examples

$f (x) = \frac{2 x - 20}{2 x^{2} + 9 x - 35}$

Step 1

Factor $2$ out of $2 x - 20$ .

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Step 1.1

Factor $2$ out of $2 x$ .

$f (x) = \frac{2 (x) - 20}{2 x^{2} + 9 x - 35}$

Step 1.2

Factor $2$ out of $- 20$ .

$f (x) = \frac{2 x + 2 \cdot - 10}{2 x^{2} + 9 x - 35}$

Step 1.3

Factor $2$ out of $2 x + 2 \cdot - 10$ .

$f (x) = \frac{2 (x - 10)}{2 x^{2} + 9 x - 35}$

Step 2

Factor by grouping.

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Step 2.1

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 $a \cdot c = 2 \cdot - 35 = - 70$ and whose sum is $b = 9$ .

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Step 2.1.1

Factor $9$ out of $9 x$ .

$f (x) = \frac{2 (x - 10)}{2 x^{2} + 9 (x) - 35}$

Step 2.1.2

Rewrite $9$ as $- 5$ plus $14$

$f (x) = \frac{2 (x - 10)}{2 x^{2} + (- 5 + 14) x - 35}$

Step 2.1.3

Apply the distributive property.

$f (x) = \frac{2 (x - 10)}{2 x^{2} - 5 x + 14 x - 35}$

Step 2.2

Factor out the greatest common factor from each group.

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Step 2.2.1

Group the first two terms and the last two terms.

$f (x) = \frac{2 (x - 10)}{(2 x^{2} - 5 x) + 14 x - 35}$

Step 2.2.2

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

$f (x) = \frac{2 (x - 10)}{x (2 x - 5) + 7 (2 x - 5)}$

Step 2.3

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

$f (x) = \frac{2 (x - 10)}{(2 x - 5) (x + 7)}$

Step 3

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

No holes

Step 4