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### No Difference of Squares

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Date: 02/08/97 at 18:55:27
From: Jeff & Beth Johnson
Subject: algebra problem: factoring

I am in Algebra I and we are factoring. Almost the entire class has
been having problems with this problem and no one understands it.
Could you try to help?

The original problem:  180x^2y - 108xy^2 -75x^3

Working on it, I took out the greatest common factor:

3x(60xy - 36y^2 - 25x^2)

Next I broke down the -36y^2 and the -25x^2 because they are the
difference of two perfect squares:

3x[60xy(-6y-5x)(6y+5x)]

Is this right?  If it is right, is it prime?

Jennie J.
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Date: 02/08/97 at 20:08:37
From: Doctor Toby
Subject: Re: algebra problem: factoring

Dear Jennie,

3x (60 x y - 36 y^2 - 25 x^2)

Next you looked at -36 y^2 - 25 x^2 and tried to factor it as a
difference of squares. But this is *not* a difference of squares,
because -36 y^2 is not a square (unless you're getting into imaginary
numbers, which probably isn't the case here). 36 y^2 is a square,
however. Just check; (-6y - 5x) (6y + 5x) = -36 y^2 - 60 x y - 25 x^2,
which is not the same as -36 y^2 - 25 x^2.

As it turns out, factoring -36 y^2 - 25 x^2 doesn't help you that
much. Suppose, for the sake of argument, that factoring
-36 y^2 - 25 x^2 = (-6y - 5x) (6y + 5x) were correct. Then you still
wouldn't have 3x (60xy (-6y - 5x) (6y + 5x)).  Instead, you would have
3x (60xy + (-6y - 5x) (6y + 5x)).  This doesn't help much since you
need a factoring that includes the 60xy.

But you are on the right track! It *is* important that 36y^2 and 25x^2
are both squares. Since there is no difference of squares (and you
need to involve the 60xy anyway), you can't use the rule for
difference of squares. What you need to learn to factor is something
which is *itself* a square.

Look at these two products:

(A + B)^2 = A^2 + 2 A B + B^2
(A - B)^2 = A^2 - 2 A B + B^2

(You can check these for yourself by multiplying.)

Notice that there are squares in these formulas also, as well as the
*cross term* 2AB.

You have 3x (60 x y - 36 y^2 - 25 x^2). You can't use your squares
yet, because they're negative squares right now. So factor out -1 to
get -3x (-60 x y + 36 y^2 + 25 x^2), or -3x (36 y^2 - 60 x y + 25 x^).
This fits one of the patterns in the last paragraph, so you should be
able to factor the expression now.

I hoped I helped!

-Doctor Toby,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
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Associated Topics:
Middle School Factoring Expressions

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