Factoring Polynomials with Common Expressions

Date: 08/20/2005 at 13:11:00
From: Eiven
Subject: Algebra is too confusing!!!!


Simplify 8(x-3) - 2(x-3)²

  A 2(x-3)(x-3)
  B 2(x-3)(x-7)
  C 2(x-3)(7-x)
  D 2(3-x)(7-x)

This is one of the questions during a quiz. I can't seem to find 
the right answer.  Here's what I've tried:

  = 8x - 24 - 2x² + 12x + 18
  = 20x - 2x² - 6
  = 2(x² + 10x - 3)

Date: 08/20/2005 at 13:44:47
From: Doctor Ian
Subject: Re: Algebra is too confusing!!!!

Hi Eiven,

There is a hard way to do this, and an easy way.  The hard way is to
go ahead and expand everything:

    8(x-3) - 2(x-3)^2 

  = (8x - 24) - 2(x^2 - 6x + 9)

  = (8x - 24) - (2x^2 - 12x + 18)

  = 8x - 24 - 2x^2 + 12x - 18            [Note the final sign]

  = 20x - 2x^2 - 42

  = -2x^2 + 20x - 42

  = -2(x^2 - 10x + 21)

  = -2(x - 7)(x - 3)

  = 2(7 - x)(x - 3)

It looks like you were doing fine, but you missed a sign at one step.  

But here's an easier way to do it.  Suppose we replace the expression
(x-3) with a single variable, like u.  Then we have

    8u - 2u^2

  = u(8 - 2u)

  = 2u(4 - u)

Now we can undo the substitution by putting (x-3) back in for u:

  = 2(x - 3)(4 - (x - 3))

  = 2(x - 3)(7 - x)

That was a lot less work, huh?  How did I do that?  Well, whenever I
see an expression that turns up in more than one place, like (x-3),
I'll try replacing it with a single variable.  That will let me spot
patterns that are harder to see with all the expressions fully expanded. 

Does this help?

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

Date: 08/23/2005 at 03:07:05
From: Eiven
Subject: Thank you (Algebra is too confusing!!!!)

Thanks a lot. Your solution makes it look a lot easier to me.