Sun K.

asked • 02/02/14

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
 
Please show all your work.

2 Answers By Expert Tutors

By:

Steve S. answered • 02/02/14

Tutor
5 (3)

Tutoring in Precalculus, Trig, and Differential Calculus

Parviz F.

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[(
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02/02/14

Parviz F.

 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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02/02/14

Steve S.

Your function is WAY OFF; see http://www.dropbox.com/s/ltwwb1acit738e6/sks23cu14020301.png
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02/03/14

Sun K.

Thanks.
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02/03/14

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