Associated Topics || Dr. Math Home || Search Dr. Math

### 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

```

```
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/
```
Associated Topics:
High School Basic Algebra
High School Polynomials

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search