How to Factor

Date: 7/25/96 at 13:33:39
From: Anonymous
Subject: An Example and Explanation of How to Factor

Here's the problem: I'm working on this summer review packet and I 
have totally forgotten how to factor. I need help with stuff like 
8x^2+3x-5.

If you could just give me an example, I could probably get the rest 
myself. Thanks!

Date: 7/28/96 at 20:33:0
From: Doctor Erich
Subject: Re: An Example and Explanation of How to Factor

Louie,

There are a couple of ways to factor polynomials.  If you look at 
a polynomial like 8x^2+3x-5, you can notice a couple of things which 
will help you factor it.  

First of all, the coefficient of the x^2 term and the coefficient of the 
constant(x^0) term, in this case 8 and -5 , are very helpful. It tells 
you that if the polynomial is factorable it will have to look something 
like (ax + b)(cx + d) where a*c = 8 and b*d = -5.  

This helps us out because now we know that either b or d must be negative,
while a and c have to be something like 2 and 4, or -8 and -1.  Since the
middle term is a 3, you also know that a*d + b*c = 3.  

This gives enough information to try plugging in a few numbers and see 
what happens. We know either b or d are 1 and -5 or -1 and 5, so let's 
try the first set.  (ax + 1)(bx - 5)....Hmmmm.  

Now let's use the other information at our disposal. -5a + b = 3 and 
a*b = 8.  If you really get desperate, you can solve these equations 
or notice that if a = 1 and b = 8, then (x+1)(8x-5)=8x^2+3x-5.  Bingo!

If you really get stuck on any equation like ax^2+bx+c where a,b,c are 
just numbers, you can use the quadratic formula, which is a formula 
that will tell you the answers to the equation. The quadratic formula 
can be found in most textbooks. (I'd write it here, but I really don't 
think I could do it justice on this computer.)  It's really helpful, 
especially if you get a really nasty polynomial... but remember it 
only works if x^2 is the largest power you're raising something to.  

If you have to deal with bigger polynomials, something like
x^4 + 10x^3 + 8x^2 - 15, you have to use factor division, which is in 
most high school Algebra II books.  If you've never seen a problem 
like the one I wrote down, don't worry about factor division; you 
might learn it next year in school.  

Good luck in your math classes next year in school.

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