Question

f(x) = (x^3+5x^2-28x-32)/(x^3-18x^2+105x-196)
which can be written as
f(x) = [(x+1)(x-4)(x+8)]/[(x-7)^2(x-4)

The domain of the function f(x), in interval notation from left to right, is ____ U ____ U ______.

The root(s) of f(x), in increasing order, is/are: _____, ____.

f(x) has hole(s) at the point(s): (____,____).

f(x) has vertical asymptotes when x is: ___.

f(x) has a horizontal asymptote at y = ___.

Answer 1

The domain of the function is all Real, except for those two points which will result in a zero in the denominator, and hence f(x) will not be defined. One of these points is x=7.
The roots can be deduced from the numerator of f(x). One of the roots is x+1=0, or x=-1.
Vertical asymptotes occur where the denominator become zero.
Horizontal asymptote is the limit of the function when x->-&inf; or x->+&inf;.
In this case, both converge to y=1.

Answer 2

Vertical asymptote at x = 7 , and a "hole" at (4, 20/3)