Factoring Equations with Many Variables

Date: 12/12/95 at 21:37:35
From: Anonymous
Subject: algebra

Hello again! Two questions, today.

                       2
XY ( X - Y )+3Y (X - Y)

=(2X - Y )(2X - Y +3Y)
=2 (2X - Y) (X + Y)

X - Y + A (Y - X )
=X - Y - A (X -Y )
= and then what do I do next?

Thanks

Date: 12/23/95 at 22:54:22
From: Doctor Elise
Subject: Re: algebra

Hi!

In algebra with polynomials like this you have to be very 
careful and thorough, because it's really easy to lose track of 
X's and Y's (they're sneaky!).  

We have a couple of options for the first one:

                             2
We can take xy(x-y) + 3y(x-y) and factor out the (x-y) part 
to get

(x-y)(xy + 3y(x-y)) and expand it to 

                   2
(x-y)(xy + 3xy - 3y ) and add up the xy parts to get

              2
(x-y)(4xy - 3y )

Or, we can try to multiply EVERYTHING out, and get

                 2   2      2      2         2
xy(x-y) + 3y(x-y) = x y - xy + 3y(x - 2xy + y )

which expands again to:

 2      2     2       2     3   
x y - xy  + 3x y - 6xy  + 3y    

                2                  2
and add up the x y parts and the xy  parts, for:

  2       2     3
4x y - 7xy  + 3y

>                       2
>XY ( X - Y )+3Y (X - Y)
>
>=(2X - Y )(2X - Y +3Y)   

(I think you'd better check this again!)

>=2 (2X - Y) (X + Y)>

Second question:

>X - Y + A (Y - X )
>=X - Y - A (X -Y )
>= and then what do I do next?

Good start!  You've noticed that there is a common "X - Y" 
thing going on here.  

Try writing it like this:

1(x - y) - A(x - y)

Can you figure it out now?

-Doctor Elise,  The Math Forum