# How do you identify all asymptotes or holes for y=(4x^3+32)/(x+2)?

Nov 3, 2016

The function is a parabola

#### Explanation:

Let's do the long division
$\textcolor{w h i t e}{a a a a}$$4 {x}^{3}$$\textcolor{w h i t e}{a a a a a a a a a a a}$$+ 32$∣$x + 2$
$\textcolor{w h i t e}{a a a a}$$4 {x}^{3} + 8 {x}^{2}$$\textcolor{w h i t e}{a a a a a a a a a}$∣$4 {x}^{2} - 8 x + 16$
$\textcolor{w h i t e}{a a a a a a}$$0 - 8 {x}^{2}$
$\textcolor{w h i t e}{a a a a a a a a}$$- 8 {x}^{2} - 16 x$
$\textcolor{w h i t e}{a a a a a a a a}$$- 8 {x}^{2} - 16 x$
$\textcolor{w h i t e}{a a a a a a a a a a a a}$$0 + 16 x + 32$
$\textcolor{w h i t e}{a a a a a a a a a a a a a a}$$+ 16 x + 32$
$\textcolor{w h i t e}{a a a a a a a a a a a a a a a a}$$+ 0 + 0$

The result is $\frac{4 {x}^{3} + 32}{x + 2} = 4 {x}^{2} - 8 x + 16$
$= 4 \left({x}^{2} - 2 x + 4\right) = 4 {\left(x - 2\right)}^{2}$
This is a parabola
graph{4(x-2)^2 [-25.65, 25.67, -12.83, 12.84]}