How many zeros does p(x) have?

Question

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.

Answer 1

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.

Here's a sketch: http://www.dropbox.com/s/ggldr45d20mhai8/sks23cu2max1min2realZeros.png

 

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.

]

Answer 2