Factoring Expressions

Date: 01/19/97 at 14:12:30
From: Heather
Subject: Algebra factoring

I have four questions that I have no idea how to answer.

Factor:

(1) 3x - 21
(2) 5x^2y-15xy^2
(3) 18x^2 - 27x
(4) 2a - 8b - 10

Please help.

Date: 01/19/97 at 16:09:57
From: Doctor Toby
Subject: Re: Algebra factoring

The first thing you do when you want to factor an expression is to 
look at the coefficients of each term. If these coefficients have a 
common *numerical* factor, you can factor that out of the entire 
expression. For example, in 3x - 21, the coefficients are 3 and -21.
These numbers are both multiples of 3, so you can factor 3 out. Then 
3x - 21 becomes 3(x - 7).  

Each of the expressions you gave has some numerical factor which can 
be factored out in this way. Do that now.

Now look back at the first problem, 3x - 21 = 3(x - 7). Can you 
factor x - 7 further?  The answer is no. Whenever you have a factor 
of the form <variable> + <number>, that factor cannot be factored any 
more.  Ideally, you want to factor every expression into the form
<number>(<variable> + <number>)...(<variable> + <number>). That's 
exactly what you have with 3(x - 7). So there's nothing more to do 
with that problem; it's finished.

Now look at the last problem, 2a - 8b - 10. It should look like 
2(a - 4b - 5) right now. From the point of view of the variable a,
-4b - 5 could just as well be a number, because the variable b has 
nothing to do with a. So 2(a - 4b - 5) is effectively like 
2(a + <number>), so that problem is also finished now.

Now look at the next to last problem, 18x^2 - 27x. It should look 
like 9(2x^2 - 3x) now. This problem is not solved yet; you can factor 
2x^2 - 3x further. Look at the terms 2x^2 and -3x. Since x^2 = xx, 
these terms are 2xx and -3x. In other words, there both of the form 
<something>x. So you can factor out the x. Now you have 9x(2x - 3).
Check to see that this can't be factored any more.

There is one problem left, 5x^2y - 15xy^2. This problem is the 
hardest one, and I'll leave it to you to figure out. It's like the 
problem 18x^2 - 27x in that you can factor out an x. But you can also 
factor out the variable y. And, of course, you've already factored 
out a number. Once you factor out all these things, this problem will 
be solved too.

In general, when you want to factor an expression, look at each term 
in the expression and try to find something that's in every term.  
When you find such a thing, you can factor it out of the entire 
expression. This technique is enough to factor most simple 
expressions, including the four you have today.

-Doctor Toby,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/