Introduction to Algebraic Fractions

Date: 05/29/2008 at 06:45:49
From: Autumn
Subject: Algebraic fractions and its problems.

What are algebraic fractions?  My teacher explained this in school but
I still do not understand what algebraic fractions are.  Can you help me?

Date: 05/29/2008 at 08:48:35
From: Doctor Ian
Subject: Re: Algebraic fractions and its problems.

Hi Autumn,

A fraction is just a division we haven't done yet.  So, for example,
3/4 means "3 divided by 4, whatever that would be". 

If we have algebraic expressions in place of constants, like 

  x/3

or

  (x^2 - 4)/(x + 2)

then we have an "algebraic fraction". 

But that's not a big deal.  They're just like any other fractions,
i.e., they follow the same rules.  For example, if we have a fraction
like 6/15, we can reduce it by identifying common factors in the
numerator and denominator:

   6   3 * 2   2
  -- = ----- = -          because the 3's cancel
  15   3 * 5   5

Does that look familiar?  We can do the same thing with a fraction like 

   6x^3   3 * 2 * x * x * x   2x
  ----- = ----------------- = --
  15x^2   3 * 5 * x * x        5

There's only one important difference here.  Recall that we can't
divide by zero.  That's not a problem with a fraction like 6/15, where
we know all the values... but with a fraction like (6x^3)/(15x^2), we
have to keep track of the fact that x can't ever be equal to zero. 
Even if the simplified form (2x)/5 would allow that, the original form
wouldn't. 

With quadratic and higher polynomials, we can get these kinds of
cancellations by factoring them into products of binomials, e.g.,

    x^2 - 4      (x + 2)(x - 2)   x - 2
  ------------ = -------------- = -----
  x^2 + 5x + 6   (x + 2)(x + 3)   x + 3

Again, we need to keep in mind that BOTH -2 and -3 are prohibited
values for x in the original expression. 

Does this make sense? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 05/29/2008 at 09:48:11
From: Autumn
Subject: Thank you (Algebraic fractions and its problems.)

Thank you for that. It really did help and i can manage my homework 
better! Thanks again.