Algebra Examples

$f (x) = \frac{x^{2} + 10 x + 24}{x^{2} - 10 x + 9}$

Step 1

Factor $x^{2} + 10 x + 24$ using the AC method.

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Step 1.1

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 $24$ and whose sum is $10$ .

$4, 6$

Step 1.2

Write the factored form using these integers.

$f (x) = \frac{(x + 4) (x + 6)}{x^{2} - 10 x + 9}$

Step 2

Factor $x^{2} - 10 x + 9$ using the AC method.

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Step 2.1

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 $9$ and whose sum is $- 10$ .

$- 9, - 1$

Step 2.2

Write the factored form using these integers.

$f (x) = \frac{(x + 4) (x + 6)}{(x - 9) (x - 1)}$

Step 3

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

No holes

Step 4