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April 16th, 2014, 09:53 PM   #1
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is this a polynomial equation?

It was my understanding that polynomial equations could not be divided by variables. But i came across this problem in rational expressions where supposedly one polynomial is divided by another. aren't a and b variables?

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April 16th, 2014, 10:49 PM   #2
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Originally Posted by fsswim1 View Post
It was my understanding that polynomial equations could not be divided by variables.
Why not?
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April 16th, 2014, 11:35 PM   #3
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Well in this article mentions that polynomials can have variables, constans and exponents. but not fractional exponents or negative exponents and they cant' be divided by variables. I'm confused. :\

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April 16th, 2014, 11:59 PM   #4
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I don’t see what there is to be confused about. That page is simply saying that a polynomial cannot contain terms like $\dfrac2x$ (i.e. negative powers of the variable). However, if you are given a polynomial, there’s nothing to stop you from doing what you like with it. In particular you can divide a polynomial by another polynomial other than the zero polynomial to get what’s called a rational function.

A rational function is an expression of the form $\displaystyle \frac{f(x)}{g(x)}$ where $f(x)$, $g(x)$ are polynomials with $g(x)\not\equiv0$. In abstract algebra the set of all rational functions with real coefficients is constructed as the field of fractions of the polynomial ring $\mathbb R[x]$.
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April 17th, 2014, 01:02 AM   #5
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I just thought that if you had a variable as the denominator it wasn't a polynomial. When i saw this exercise i thought " ok one polynomial over another polynomial. The first one has an a-b as denominator and the second has a 2a as denominator. a and b are both variables, therefore they can't be polynomials right?" Does it matter that the numerators are variables too?
Btw how to you insert those neat functions like f(x)g(x)? Please teach me
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April 17th, 2014, 02:34 AM   #6
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Originally Posted by fsswim1 View Post
I just thought that if you had a variable as the denominator it wasn't a polynomial.
That’s right. Rational functions are not polynomials (unless the denominator is a constant*). In other words:
  1. $f(x)$ is a polynomial,
  2. $g(x)$ is a polynomial, but
  3. the whole thing $\displaystyle \frac{f(x)}{g(x)}$ is not a polynomial.
*Or more generally, unless the denominator is a factor of the numerator.

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Originally Posted by fsswim1 View Post
Btw how to you insert those neat functions like f(x)g(x)? Please teach me
If you browse the LaTeX Help section you might be able to find a few threads with instructions on how to insert $\LaTeX$.
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Last edited by Olinguito; April 17th, 2014 at 02:37 AM.
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April 17th, 2014, 05:24 PM   #7
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Will do Thx Olinguito
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