Answer:
![\frac{4(x - 5)(x + 7)(x - 12)}{(x + 1)(x)(x - 12)}](https://tex.z-dn.net/?f=%20%5Cfrac%7B4%28x%20-%205%29%28x%20%2B%207%29%28x%20-%2012%29%7D%7B%28x%20%2B%201%29%28x%29%28x%20-%2012%29%7D%20)
Step-by-step explanation:
A rational function is
![\frac{p(x)}{q(x)}](https://tex.z-dn.net/?f=%20%5Cfrac%7Bp%28x%29%7D%7Bq%28x%29%7D%20)
where q(x) doesn't equal zero.
If p is a asymptote, or hole at that value, then we will use
![(x - p)](https://tex.z-dn.net/?f=%28x%20-%20p%29)
Step 1: We have asymptote as 0 and -1 so our denomiator will include
![(x - 0)(x - ( - 1)](https://tex.z-dn.net/?f=%28x%20-%200%29%28x%20-%20%28%20-%201%29)
Which is
![(x)(x + 1)](https://tex.z-dn.net/?f=%28x%29%28x%20%2B%201%29)
So our denomator so far is
![\frac{p(x)}{x(x + 1)}](https://tex.z-dn.net/?f=%20%5Cfrac%7Bp%28x%29%7D%7Bx%28x%20%2B%201%29%7D%20)
Step 2: Find Holes.
Since 12 is the value of the hole,
![(x - 12)](https://tex.z-dn.net/?f=%28x%20-%2012%29)
is a the binomial.
This will be both on the numerator and denomator so qe have
![\frac{(x - 12)}{x(x + 1)(x - 12)}](https://tex.z-dn.net/?f=%20%5Cfrac%7B%28x%20-%2012%29%7D%7Bx%28x%20%2B%201%29%28x%20-%2012%29%7D%20)
Step 3: Put the x intercepts in the numerator.
Since 5 and -7 is the intercepts,
![\frac{(x - 12)(x - 5)(x + 7)}{x(x + 1)(x - 12)}](https://tex.z-dn.net/?f=%20%5Cfrac%7B%28x%20-%2012%29%28x%20-%205%29%28x%20%2B%207%29%7D%7Bx%28x%20%2B%201%29%28x%20-%2012%29%7D%20)
Step 4: Horinzontal Asymptotes,
Multiply the numerator and denomiator out fully,
![\frac{ {x}^{3} - 10 {x}^{2} - 59x + 420 }{ {x}^{3} - 12 {x}^{2} + x - 12}](https://tex.z-dn.net/?f=%20%5Cfrac%7B%20%20%7Bx%7D%5E%7B3%7D%20-%2010%20%7Bx%7D%5E%7B2%7D%20%20-%2059x%20%2B%20420%20%7D%7B%20%7Bx%7D%5E%7B3%7D%20-%2012%20%7Bx%7D%5E%7B2%7D%20%20%20%2B%20x%20-%2012%7D%20)
Take a L
look at the coefficients,
Notice they have the same degree,3, this means if we divide the leading coefficents, we will get our horinzonral asymptote.
Multiply the numerator by 4.
![\frac{4(x - 12)(x - 5)(x - 7)}{x(x + 1)(x - 12)}](https://tex.z-dn.net/?f=%20%5Cfrac%7B4%28x%20-%2012%29%28x%20-%205%29%28x%20-%207%29%7D%7Bx%28x%20%2B%201%29%28x%20-%2012%29%7D%20)
Above is the function,