Regrouping Polynomials

Date: 11/26/2001 at 01:39:19
From: jessica
Subject: Elementary algebra 

I'm stuck on how to do the regrouping of polynomials. The problem is: 

   a^2x - bx - a^2y + by + a^2z - bz  

Please help. Thank you. 
Jess

Date: 11/26/2001 at 10:12:04
From: Doctor Ian
Subject: Re: Elementary algebra 

Hi Jessica,

The basic idea in regrouping is to look for opportunities to apply the 
distributive property:

    this*bibbity + this*bobbity + ... + this*boo

  = this(bibbibty + bobbity + ... + boo)

Sometimes applying it once is enough; sometimes you have to apply it 
several times.  That's the case with

   a^2x - bx - a^2y + by + a^2z - bz 

We might begin by noticing that the coefficients a^2 and b are shared 
by several terms, so we can group those together and distribute the 
multiplications:
 
  a^2x - bx - a^2y + by + a^2z - bz 

  = a^2x - a^2y + a^2z - bx + by - bz

  = a^2(x - y + z) - b(x - y + z)

  = (a^2 - b)(x - y + z) 

On the other hand, we might concentrate on the variables x and y 
instead:

  a^2x - bx - a^2y + by + a^2z - bz 

  = (a^2 - b)x - (a^2 - b)y + (a^2 - b)z

  = (a^2 - b)(x - y + z)

Either way, we end up with the same thing. What makes this problem 
tricky is that it has lots of minus signs. If you find minus signs 
confusing, you can try changing '- something' to '+ (-1)something' 
wherever it occurs. Then you only have to deal with additions:

   
  a^2x - bx - a^2y + by + a^2z - bz 

  = a^2x + (-1)bx + (-1)a^2y + by + a^2z + (-1)bz

  = a^2(x + (-1)y + z) + b((-1)x + y + (-1)z)

  = a^2(x - y + z) + b(-x + y - z)

Note that in this form, you can't use the distributive property a 
second time. So it's useful to know that you can distribute a -1 
across a sum by changing all the signs. For example, 

  (-1)(-x + y - z) = ((-1)(-x) + (-1)y + (-1)(-z))

                   = (x - y + z)

So we can use this trick to change (-x + y - z) to (x - y + z), by 
subtracting the one instead of adding the other:

  = a^2(x - y + z) + b(-x + y - z)

  = a^2(x - y + z) - b( x - y + z)

  = (a^2 - b)(x - y + z)

Once you get enough practice, switching the signs will seem like a 
very natural thing. But until then, it's probably best to write all 
the steps out explicitly, e.g., 

  = a^2(x - y + z) + b(-x + y - z)

  = a^2(x - y + z) - (-1)b(-x + y - z)
                     \_______________/
                     
                     b(-1)(-x + y - z)
                     \_______________/

                     b((-1)(-x) + (-1)(y) - (-1)(z))
                     \_____________________________/
                     
                     b(x - y + z)

  = a^2(x - y + z) - b(x - y + z)

This trick (switching signs by subtracting instead of adding) turns 
out to be useful in lots of simplification problems. One place to look 
for it is when trying to cancel common expressions in the numerator 
and denominator of a fraction, e.g., 

  -x - y   (-1)x + (-1)y   (-1)(x + y)
  ------ = ------------- = ----------- = -1
   x + y       x + y          x + y

Does this help?

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