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Let a > 0, and let P(x) be a polynomial with integer coefficients such that, P(1) = P(3) = P(5) = P(7) = a and P(2) = P(4) = P(6) = P(8) = -a. What is the smallest possible value of a?

 Sep 12, 2021
 #1
avatar
-1

The smallest possible value of a is 945.

 Sep 12, 2021
 #2
avatar+114496 
+1

I have no idea how to do this.  So if a mathematician wants to explain it that would be really good.

 

Of course, if the coefficients did not have to be integers then a could be as close to 0 as you please.

 

I did construct a graph of degree 7 but it is not actually helpful

 

Here is a link:

https://www.geogebra.org/classic/tbnjw62b

 Sep 14, 2021
 #3
avatar+32557 
+2

Using Mathcad to do the heavy lifting I get the following:

 

I suspect there is a more elegant way, but I can;t think of one!

 Sep 14, 2021
 #4
avatar+114496 
+1

Thanks Alan,

 

That works  :)     

Guest's answer is exactly 3 times more than yours.  That suggests more than just a coincidence....  

 

 

 

 

 Sep 14, 2021

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