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Sketching a Polynomial

Date: 04/04/2002 at 23:53:03
From: Shikhar 
Subject: Wavy curve method for solving inequalities 

Dear Dr. Math,

When solving inequalities of the form P(X)/Q(X) > 0 or P(X)/Q(X) < 0, 
we have to find the roots of P(X) and Q(X) and plot the roots on the 
real line in increasing order (excluding common roots). After that a 
curve is drawn starting beyond the extreme right root along the real 
line and going toward the left. This curve is said to be changing its 
position alternately between consecutive roots. 

   1. It is not clear to me why the curve should change its position 
      or sign at roots. Can't the curve have a positive value for all 
      roots?

   2. Is the curve symmetrical about the real line?
 
   3. Why exclude common roots?


Date: 04/05/2002 at 16:04:25
From: Doctor Peterson
Subject: Re: Wavy curve method for solving inequalities 

Hi, Shikhar.

I am not familiar with this method, but it makes sense to me, with 
some clarifications. Essentially, you are sketching the polynomial PQ, 
whose sign is the same as P/Q for any given x, paying attention only 
to where it is positive and negative, not to how far it goes from the 
axis.

Imagine the function factored:

       P(x)  (x - p1)(x - p2)...(x - pm)(product of quadratic factors)
f(x) = ---- = --------------------------------------------------------
       Q(x)  (x - q1)(x - q2)...(x - qn)(product of quadratic factors)

The quadratic factors, which can't be factored further, will each be 
always positive (or always negative - your description leaves out the 
fact that the highest degree terms of P and Q must be positive in 
order for your curve to start above the axis at the right). So we can 
ignore them as far as the sign is concerned. The p1 ... pm and 
q1 ... qn are the roots of P and Q.

Now, as you go along the x axis, at some point you will pass a root, 
say pk. As you do so - if there is only one factor (x - pk) - that one 
factor will change sign, and no others. Therefore, f(x) will change 
sign there. But if there are two identical factors (either in P or Q), 
they will both change sign, and f(x) will not. However, you really 
shouldn't completely ignore them, because they will cause f(x) to be 
zero (for a root of P) or undefined (for a root of Q), which will 
affect your solution. Instead, you should show your curve just 
touching the axis without crossing it.

Therefore, you have to not only cancel common factors (ignoring common 
roots), but also "cancel" pairs of identical roots within either 
polynomial. So if a root appears an even number of times in P and Q 
together, it will not show up as a crossing on your curve (though it 
will be a zero or undefined point); if it appears an odd number of 
times, it will act as if it appeared once.

You should try actually graphing several functions like this to see 
how they compare to the curve you are sketching. You will find that 
roots shared by P and Q have no effect at all (other than forming a 
"hole discontinuity" in f(x), since the function is not defined 
there); roots appearing twice in the factorization of P will make the 
curve tangent to the axis; roots of Q will produce a vertical 
asymptote (undefined value), and for double roots of Q the curve will 
have the same sign on both sides of the asymptote.

All in all, I think I would rather see you learning to sketch rational 
functions (as I just described in the last paragraph) and apply the 
result to the inequality, rather than learn a related, but less 
accurate, trick for handling the inequality alone. Curve sketching 
gives you a fuller understanding of how the function behaves, and is 
not much more work than what you have been taught.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School Equations, Graphs, Translations
High School Functions

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