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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 in g(x) with even exponents. (For example, the sum of the coefficients of the terms in -7x^3+4x^2+10x-5 with even exponents is 4+(-5)=-1). 

 

*I've already found a, b, and c and I just need to find a way to find the coefficients of even terms and add them up without doing ^16.

 

Thx

 Feb 5, 2021
 #1
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If your example

\(\displaystyle -7x^{3}+4x^{2}+10x-5=h(x)\quad \text{say,} \\ \text{then} \\ h(1)+h(-1) \\=(-7+4+10-5)+(7+4-10-5)=2-4=-2=2(-1).\)

Even powered terms double up, odd powered terms cancel.

 Feb 5, 2021
 #2
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So then if I did like f(x)=y0+x1y1+...x32y32 then I use x=1=-1 and add and I get f(1)+f(-1)=2(y0+y2+y4) and since I've already found a, b, and c, i can put that in a normal quadratic polynomial and I have my answer?

Guest Feb 5, 2021

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