General Sign Patterns for Factoring Trinomials

Date: 04/12/2005 at 22:51:44
From: Chris
Subject: Signs in Trinomials

How can you tell whether the numbers should be negative or positive 
in trinomials?  I just don't seem to get it.  Can you help me?

Date: 04/13/2005 at 10:40:55
From: Doctor Ian
Subject: Re: Signs in Trinomials 

Hi Chris,

You can tell by looking at the pattern.  There are three 
possibilities:

  1. (x + a)(x + b)        Both added

  2. (x + a)(x - b)        One added, one subtracted

  3. (x - a)(x - b)        Both subtracted

(Assume that a and b are both positive.) 

By expanding these, we can see that the pattern of signs in the 
quadratic tells us which one we're dealing with:

  1. (x + a)(x + b) = x^2 + (a+b)x + ab         

  2. (x + a)(x - b) = x^2 + (a-b)x - ab

  3. (x - a)(x - b) = x^2 - (a+b)x + ab

So 

  x^2  +  __x + __     ->   Case 1

  x^2  -  __x + __     ->   Case 3

  x^2 +/- __x - __     ->   Case 2

In practice, I haven't actually memorized the pattern.  What I do is
look at the last sign, and if it's negative, I know that I have to
have one added and one subtracted:

   ... - 3   ->  (x + a)(x - b)

Otherwise, I look at the second sign, and if it's negative, I know I
have to have two subtractions:

   ... - 5 + ...    ->   (x - a)(x - b)

Otherwise, I know I have two additions. 

Does this make sense?  

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/