(4x^2+ 4x -24) / (x^4- 2x^3 - 9x^2 +18x)
-
Vertical Asymptotes are where the equation does not exist. Since it's a fraction, its when the denominator equals zero. However, check the numerator also to see if any holes are in the graph.
x^4 - 2x^3 -9x^2 + 18x
x (x^3 - 2x^2 - 9x + 18)
x ( x^2(x-2) - 9 (x-2))
Factor by grouping
x (x^2-9) (x-2)
Factor x^2-9 as a difference of squares
x (x-3) (x+3) (x-2)
Numerator factoring:
4 (x^2 + 2x - 6)
4(x+4)(x-2)
The entire fraction is:
4(x+4)(x-2)
-------------------
x (x-3) (x+3) (x-2)
Since (x-2) cancels on the numerator/denominator, it is a hole (not a vertical asymptote) in the graph.
Therefore, the VA's are: x = 0, x = 3, x = -3
x^4 - 2x^3 -9x^2 + 18x
x (x^3 - 2x^2 - 9x + 18)
x ( x^2(x-2) - 9 (x-2))
Factor by grouping
x (x^2-9) (x-2)
Factor x^2-9 as a difference of squares
x (x-3) (x+3) (x-2)
Numerator factoring:
4 (x^2 + 2x - 6)
4(x+4)(x-2)
The entire fraction is:
4(x+4)(x-2)
-------------------
x (x-3) (x+3) (x-2)
Since (x-2) cancels on the numerator/denominator, it is a hole (not a vertical asymptote) in the graph.
Therefore, the VA's are: x = 0, x = 3, x = -3
-
Whatever x values will make the denominator zero are the vertical asymptotes