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How many zeros does p(x) have?

A polynomial p(x) has a relative maximum at (-2, 4), a relative minimum at (1, 1), a relative maximum at (5, 7) and no other critical points. How many zeros does p(x) have? 
 
Answer: Two
 
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2 Answers

relative maximum at A(-2, 4)
 
relative minimum at B(1, 1)
 
relative maximum at C(5, 7)
 
               |           /•\ C
        A     |     
       /•\    |
               |  \•/ B
------------|-------------
               |
 
With no other critical points and 3 turning points the degree is 4 and there are 4 zeros, but only two will be real because the graph will cross the x-axis in only 2 points.
 
CORRECTION: There could be 4 + 2n, n integer, zeros; but only 2 will be real. I found that using a degree 4 p(x) wouldn't go through all three points. So I changed my p'(x) to p’(x) = a(x+2)(x-1)(x-5)((x-b)^2+c^2) which made the degree of p(x) six, with 2 real zeros and 4 imaginary zeros.
 
REDACTED:
[
Or, using calculus, p’(x) = a(x+2)(x-1)(x-5). which has degree 3, so the degree of p(x) = 3 + 1 = 4.

4th degree ==> 4 zeros; two are real x-intercepts.
 
Here's a GeoGebra curve fit to the given points and to two points on the x-axis: http://www.dropbox.com/s/pfhbvq63h2r02lc/sks23cu14020302.png. Revised 2/4/14; check it out.
 
GeoGebra gives the numeric coefficients for a 4th degree polynomial function, p(x), and a 3rd degree polynomial function for the derivative of p(x), p'(x).
 
p(x) IS DEGREE 4, p'(x) IS DEGREE 3.
]

Comments

Hello Steve:
  Only way P' ( x) has a factor of ( X+2) if P( X) has factor of ( X +2) ^2 and relative minimum will be at (-2,0) , which is not the case here.
   Relative maximum of ( -2 , 4 ) , implies that
 
  P(X ) = k[(
 P(x ) has factor of a quadratic with roots conjugate pairs of complex numbers, and accordingly each relative minimum and maximum counts for factor of quadratic with conjugate pairs of complex numbers as their roots, and also 1 real root X < -2. 
P(X) = a [ ( X+2) ^2 - 4) ] [ ( X - 1) ^2 +1 ] [ (X -5) ^2 + 7]
 
 Complex roots come into conjugate pairs , so 3 pairs adds up to 6 roots , and one real root for X < -2
 
   So, degree of polynomial is 7 , 3 pairs of conjugate complex roots , and 1 rational root = 7 roots equal the degree of polynomial 7.
 
 

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