Question

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

Answer 1

The function is a parabola

Explanation:

Let's do the long division
`color(white)(aaaa)` `4x^3` `color(white)(aaaaaaaaaaa)` `+32` `∣` `x+2`
`color(white)(aaaa)` `4x^3+8x^2` `color(white)(aaaaaaaaa)` `∣` `4x^2-8x+16`
`color(white)(aaaaaa)` `0-8x^2`
`color(white)(aaaaaaaa)` `-8x^2-16x`
`color(white)(aaaaaaaa)` `-8x^2-16x`
`color(white)(aaaaaaaaaaaa)` `0+16x+32`
`color(white)(aaaaaaaaaaaaaa)` `+16x+32`
`color(white)(aaaaaaaaaaaaaaaa)` `+0+0`

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