# Thread: Distinguishing between a polynomial and rational function

1. ## Distinguishing between a polynomial and rational function

Greetings M.H.F.

I am learning to distinguish between rational and polynomial functions. A questions reads:

"State whether the function is a polynomial, a rational function(but
not a polynomial), or neither a polynomial nor a rational function. If
the function is a polynomial, give the degree."

g(x)= [x^2 - 2x - 8] / [x + 2]

I know that the top tier is a polynomial of degree 2 and the bottom tier is a polynomial of degree 1. By definition, a rational function is a polynomial divided by a polynomial where the lower tier cannot equal zero, which is what I seem to have here.

Since the function is rational, I do not have to give the degree, but can rational functions have a degree? Yes or no? If so, how would I calculate it.

Thanks,
Jack Daniels

2. Originally Posted by UC151CPR
Greetings M.H.F.

I am learning to distinguish between rational and polynomial functions. A questions reads:

"State whether the function is a polynomial, a rational function(but
not a polynomial), or neither a polynomial nor a rational function. If
the function is a polynomial, give the degree."

g(x)= [x^2 - 2x - 8] / [x + 2]

I know that the top tier is a polynomial of degree 2 and the bottom tier is a polynomial of degree 1. By definition, a rational function is a polynomial divided by a polynomial where the lower tier cannot equal zero, which is what I seem to have here.

Since the function is rational, I do not have to give the degree, but can rational functions have a degree? Yes or no? If so, how would I calculate it.

Thanks,
Jack Daniels

Hi Jack,

You should first note that $\displaystyle x^2-2x-8 = (x+2)(x-4)$

This gives us that $\displaystyle \frac{x^2-2x-8}{x+2} = \frac{(x+2)(x-4)}{x+2} = x-4$ and so this is actually a polynomial of degree one.

3. ## Are you sure?

Not to question your knowledge, but after doing research I came accross a site:

Rational Functions

And it has an example of a rational function being:

f(x) = (2x + 2) / (x + 1)

It says that despite the (X+1)'s crossing off, that it is still a rational function, just with a hole at x=-1.

Do you have any examples online supporting your claim that it is a polynomial?

4. Originally Posted by UC151CPR
Not to question your knowledge, but after doing research I came accross a site:

Rational Functions

And it has an example of a rational function being:

f(x) = (2x + 2) / (x + 1)

It says that despite the (X+1)'s crossing off, that it is still a rational function, just with a hole at x=-1.

Do you have any examples online supporting your claim that it is a polynomial?
Oops, you're right! :P I should have been more cautious.

As for your other question, there isn't really a definition for the degree of a rational function, so it really only depends on what you make out of it. You could say that the degree is the highest degree of the numerator minus the highest degree of the denominator, for example, but I don't think it is defined anywhere nor that it really matters..