Sign Diagrams

Date: 05/20/2002 at 03:43:17
From: Jessie
Subject: Sign Diagrams (without graphing)

There is a diagram in my text book that looks something like this:

   x2 - 1/2x



           |                |
     +     |       _        |   +
    __________________________________
           |                |
         -1/2               |


I have no idea what it's telling me.  Can you help me make sense of
it? 

Date: 05/20/2002 at 12:24:14
From: Doctor Peterson
Subject: Re: Sign Diagrams (without graphing)

Hi, Jessie.

When you have a polynomial factored, you can find the sign of the 
product by considering the sign of each factor separately. So let's 
first factor our expression as

    x^2 - 1/2 x = x(x - 1/2)

The first factor, x, will of course be positive when x > 0 and 
negative when x < 0:

    --------------0+++++++++++++  x
    <-------------+------------>
                  0

The second factor, x - 1/2, will be positive when x > 1/2:

    -------------------0++++++++  x - 1/2
    <-------------+----+------->
                  0   1/2

Each factor has divided the number line into two parts, in different 
ways. If we put these together, we make a total of three regions of 
the number line, and can see what the product of the signs will be in 
each region:

    --------------0+++++++++++++  x
    -------------------0++++++++  x - 1/2
    ++++++++++++++0----0++++++++  x(x - 1/2)
    <-------------+----+------->
                  0   1/2

Whenever both factors are positive or both are negative, the product 
will be positive; when one is positive and the other negative, the 
product will be negative. That gives us a simple way to find the sign 
of the expression everywhere.

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