Find the Holes in the Graph f(x)=(x^3-16x)/(-4x^2+4x+24)

$f (x) = \frac{x^{3} - 16 x}{- 4 x^{2} + 4 x + 24}$

Step 1

Factor $x^{3} - 16 x$ .

Factor $x$ out of $x^{3} - 16 x$ .

Factor $x$ out of $x^{3}$ .

$f (x) = \frac{x \cdot x^{2} - 16 x}{- 4 x^{2} + 4 x + 24}$

Factor $x$ out of $- 16 x$ .

$f (x) = \frac{x \cdot x^{2} + x \cdot - 16}{- 4 x^{2} + 4 x + 24}$

Factor $x$ out of $x \cdot x^{2} + x \cdot - 16$ .

$f (x) = \frac{x (x^{2} - 16)}{- 4 x^{2} + 4 x + 24}$

Rewrite $16$ as $4^{2}$ .

$f (x) = \frac{x (x^{2} - 4^{2})}{- 4 x^{2} + 4 x + 24}$

Factor.

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 4$ .

$f (x) = \frac{x ((x + 4) (x - 4))}{- 4 x^{2} + 4 x + 24}$

Remove unnecessary parentheses.

$f (x) = \frac{x (x + 4) (x - 4)}{- 4 x^{2} + 4 x + 24}$

Step 2

Factor $- 4 x^{2} + 4 x + 24$ .

Factor $4$ out of $- 4 x^{2} + 4 x + 24$ .

Factor $4$ out of $- 4 x^{2}$ .

$f (x) = \frac{x (x + 4) (x - 4)}{4 (- x^{2}) + 4 x + 24}$

Factor $4$ out of $4 x$ .

$f (x) = \frac{x (x + 4) (x - 4)}{4 (- x^{2}) + 4 (x) + 24}$

Factor $4$ out of $24$ .

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

Factor $4$ out of $4 (- x^{2}) + 4 x$ .

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

Factor $4$ out of $4 (- x^{2} + x) + 4 \cdot 6$ .

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

Factor.

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 $a \cdot c = - 1 \cdot 6 = - 6$ and whose sum is $b = 1$ .

Multiply by $1$ .

$f (x) = \frac{x (x + 4) (x - 4)}{4 (- x^{2} + 1 x + 6)}$

Rewrite $1$ as $- 2$ plus $3$

$f (x) = \frac{x (x + 4) (x - 4)}{4 (- x^{2} + (- 2 + 3) x + 6)}$

Apply the distributive property.

$f (x) = \frac{x (x + 4) (x - 4)}{4 (- x^{2} - 2 x + 3 x + 6)}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$f (x) = \frac{x (x + 4) (x - 4)}{4 ((- x^{2} - 2 x) + 3 x + 6)}$

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

$f (x) = \frac{x (x + 4) (x - 4)}{4 (x (- x - 2) - 3 (- x - 2))}$

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

$f (x) = \frac{x (x + 4) (x - 4)}{4 ((- x - 2) (x - 3))}$

Remove unnecessary parentheses.

$f (x) = \frac{x (x + 4) (x - 4)}{4 (- x - 2) (x - 3)}$

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^3-16x)/(-4x^2+4x+24)

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