Cancelling Algebraic Fractions

Date: 11/11/2002 at 07:17:15
From: Dean James
Subject: Algebraic fractions

I am currently doing some last day revision for my maths GCSE (Higher 
Tier), which I will be taking over this week.

I have looked through all of my study materials, but I still don't 
understand how to cancel down algebraic fractions, as they don't 
explain it very thoroughly.

Could you please explain?

Thanking you in anticipation,
Dean James

Date: 11/11/2002 at 10:19:39
From: Doctor Ian
Subject: Re: Algebraic fractions

Hi Dean,

It's basically the same idea as reducing other kinds of fractions
using prime factors. For example, here's how to do it using only 
numbers:

  72   2 * 2 * 2 * 3 * 3        2   2   3   2 * 3
  -- = ---------------------- = - * - * - * ----- = 6/5
  60       2 * 2 * 3     * 5    2   2   3     5

You can do exactly the same thing using variables as your 'prime 
factors':

  x^3 y^4 z     x * x * x * y * y * y * y * z
  ----------- = -----------------------------------------
  x   y^2 z^4           x * y * y         * z * z * z * z


                x * x * y * y
              = -------------
                    z * z * z


                x^2 y^2
              = -------
                  z^3

Eventually, as you get more practice with this, you'll notice that you
can accomplish the same thing just by subtracting exponents:

  x^3 y^4 z     
  ----------- = x^(3-1) y^(4-2) z^(1-4) = x^2 y^2 z^-3
  x   y^2 z^4    

The other kind of cancellation that often occurs requires you to
factor polynomials, e.g., 

  x^2 + 3x + 2   (x+1)(x+2)   (x+1)   (x+2)
  ------------ = ---------- = ----- * ----- = (x+2)/(x-1)
     x^2 - 1     (x+1)(x-1)   (x+1)   (x-1)

But it's all really the same thing: Break the numerator and 
denominator into the most basic possible factors; and eliminate any
factor that appears in both the numerator and the denominator. 

Does this help? 

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