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### 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

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.
```
Associated Topics:
High School Polynomials
Middle School Factoring Expressions

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