Math 4281      Spring 2006
Suggestions for ex 6 of Sec 15D

      This is a continuation of the discussion in class on Monday, May 1. Let's go through some of the various parts of the problem. Although some of them may look strange at first, it's possible to use methods from first-year calculus to deal with various parts.
 

1.   The suggestion here is to show that a real polynomial  f(x)  of odd degree must have a real root.
   Please recall that if  deg(f)  is odd, then  f(x) --> +  as  x --> +, and also
   f(x) --> -  as  x --> -.  Therefore  f(a) > 0  if  a is a sufficiently large positive integer,
   and  f(b) < 0  if  b is a sufficiently large negative integer. Now ¿what did we learn in calculus about a function
   that's positive at one end of an interval and negative at the other endpoint?
    
    
    
    
    
2.   Here, we can begin by writing  f(x) = g(x)²h(x),  or f(x) = g(x)^dh(x),  where  d  is an unspecified integer > 2. 
   {In the second version  d  would presumably be the actual multiplicity of the factor; however it may
   not be necessary to be this precise}. If we take the derivative of both sides, applying the various
   standard formulas of calculus, we should be able to obtain something that exhibits  g(x)  as a
   common factor of  f(x)  and  f '(x).
    
    
    
    
    
3.   We are given 3 conditions:  (1) a multiple root of  f(x)  {or  x-a as a multiple factor}, 
   (2) a statement about the graph of  y = f(x),  and  (3) a common root of  f(x)  and  f '(x). 
   The connection between (1) and (3) is related to part ii above, while the connection
   between (2) and (3) is discussed in calculus courses under the heading of "curve sketching".
    

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