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### 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/
```
Associated Topics:
Middle School Factoring Expressions

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