Expert: Steve Holleran - 7/16/2007

Question
I'm having a lot of difficulties with these three questions. Thanks for your help!

1. Is 2/3 a zero of f(x) = x7+ 6x5 – x4 + x + 2? Explain why or why not.

2. Write the equation of one rational function f(x)= p(x)/g(x)  having the indicated properties, in which the degrees of p and q are as small as possible.
a) f has a vertical asymptote given by x = 1
b) f has a slant asymptote whose equation is y = x
c) f has a y-intercept at 2
d) f has x-intercepts at -1 and 2

3. Can the graph of one rational function have both a horizontal asymptote and a slant asymptote? Explain why or why not.

I just can't seem to figure the way to go about solving these. PLease help! Thanks in advance!


Answer
Hi Jillian,

Okay, let's see what we can do here.

1.  There's a quick way to answer this one.  If you check out something called the Rational Root Theorem, it states that any rational zeros of a polynomial MUST be of the form:

    (factors of constant) / (factors of lead coefficient)

Here, the constant is 2 and the lead is 1, so the possible rational zeros are :  +/- {2/1 , 1/1} .  Therefore , 2/3 is not a possible zero.

2.  Vertical asymptotes occur where the bottom is zero and the top is not, so how about :

              f(x) = 2 / (x-1)  ?

   Slant asymptotes come up when the top is a degree higher than the bottom, and the lead coefficients of the top and bottom here have to be the same, so let's say:

             f(x) = (x^2) / (x + 1). This will have y = x as a slant asymptote as x tends to - infinity.

    Here, you want y to bee 2 when x = 0, so I think

f(x) = (x^2 + 2) / (x + 1) works out okay.

   In this last part, you want y = 0 at x = -1 and 2, so the factors of the top have to be (x+1) and (x-2), so how about:

    f(x) = (x+1)(x-2) / (x+6) = (x^2 -x -2) /(x - 6) ?


3.  No, it has to have one or the other.  For a horizontal asymptote, the degrees on the top and bottom have to be the same, and for a slant asymptote the top has to be one degree higher than the bottom, so you have to have one case or the other, but not both.

I hope these were able to help you out.

Steve Holleran