Find the Holes in the Graph f(x)=(x^2+4x+4)/(x^2+2x-8)

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

Step 1

Factor using the perfect square rule.

Rewrite $4$ as $2^{2}$ .

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

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 x = 2 \cdot x \cdot 2$

Rewrite the polynomial.

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

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 2$ .

$f (x) = \frac{{(x + 2)}^{2}}{x^{2} + 2 x - 8}$

Step 2

Factor $x^{2} + 2 x - 8$ 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 $- 8$ and whose sum is $2$ .

$- 2, 4$

Write the factored form using these integers.

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

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)=(x^2+4x+4)/(x^2+2x-8)

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