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Let f(x) be a quadratic polynomial such that f(-4)=-22, f(-1)=2, and f(2)=-1. Let g(x)=f(x)^16. Find the sum of the coefficients of the terms ing(x)  that have even degree. (For example, the sum of the coefficients of the terms in -5x^3+4+11x-5 that have even degree is (4)+(-5)=-1.)

 

I figured out that f(x)=-1.5x^2+0.5x+4, but I'm not sure how to solve the problem from there.

 Jun 8, 2020
edited by mathmathj28  Jun 8, 2020
 #1
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The sum of the coefficients is clearly (2^16 + (-1)^16)/2.= (2^16 + 1)/2.

 Jun 8, 2020
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Could you please explain how you got there? Thanks

mathmathj28  Jun 8, 2020
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I would like to see someone present this answer too.  frown

 

Your function f certainly works but I do not know what to do next either.

 Jun 8, 2020
edited by Melody  Jun 8, 2020
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Consider g(1) = f(1)^16  This will consist of the sum of the even degree coefficients and the sum of odd degree coefficients..  Now consider g(-1) = f(-1)^16.  This will consist of the sum of the even degree coefficients and the negative of the sum of the odd degree coefficients.  Add g(1) and g(-1).  This will eliminate the odd degree coefficients, but will leave twice the sum of the even degree coefficients. Hence to get the sum of the even degree coefficients you need:

          

                                                (g(1) + g(-1))/2  or  (f(1)^16  +  f(-1)^16)/2

 Jun 8, 2020
edited by Alan  Jun 8, 2020
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Thanks very much Alan,

It took me a while to work out what you were saying but I have it all sorted now.

I never would have thought to do that!  

 

Have you worked it out mathmathj28?

 

If so what answer did you finally get?

Melody  Jun 8, 2020
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Thank you so much for the help Alan!! I didn't think of doing that before but I understand why that is helpful now.

 

If my work is correct, the sum of the coefficients of the terms in g(x) that have even degree is 21,556,128.5

\(g(1)=f(1)^{16}=(-\frac{3}{2}+\frac{1}{2}+4)^{16}=3^{16}=43,046,721 \)

\(g(-1)=f(-1)^{16}=(-\frac{3}{2}-\frac{1}{2}+4)^{16}=2^{16}=65,536\)

\(\frac{g(1)+g(-1)}{2}=\frac{43,046,721+65,536}{2}=\frac{43,112,257}{2}=21,556,128.5\)

 

Thanks again for helping me solve this!

mathmathj28  Jun 8, 2020

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