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mukushla
Group Title
Suppose \(f(x)\) is a degree \(8\) polynomial such that \(f(2^i)=\frac{1}{2^i}\) for all integers \(0≤i≤8\). Evaluate \(f(0)\).
 one year ago
 one year ago
mukushla Group Title
Suppose \(f(x)\) is a degree \(8\) polynomial such that \(f(2^i)=\frac{1}{2^i}\) for all integers \(0≤i≤8\). Evaluate \(f(0)\).
 one year ago
 one year ago

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Mendicant_Bias Group TitleBest ResponseYou've already chosen the best response.0
The text is a little small, what is two to the power of in f(2^[this])?
 one year ago

mukushla Group TitleBest ResponseYou've already chosen the best response.0
\[\large f(2^i)=\frac{1}{2^i}\]its \(i\)
 one year ago

Mendicant_Bias Group TitleBest ResponseYou've already chosen the best response.0
Thanks.
 one year ago

amistre64 Group TitleBest ResponseYou've already chosen the best response.0
i would setup a matrix to row reduce
 one year ago

amistre64 Group TitleBest ResponseYou've already chosen the best response.0
rref{{0^8,0^7,0^6,0^5,0^4,0^3,0^2,0,1,1/2^0}, {1^8,1^7,1^6,1^5,1^4,1^3,1^2,1,1,1/2^1}, {2^8,2^7,2^6,2^5,2^4,2^3,2^2,2,1,1/2^2}, {3^8,3^7,3^6,3^5,3^4,3^3,3^2,3,1,1/2^3}, {4^8,4^7,4^6,4^5,4^4,4^3,4^2,4,1,1/2^4}, {5^8,5^7,5^6,5^5,5^4,5^3,5^2,5,1,1/2^5}, {6^8,6^7,6^6,6^5,6^4,6^3,6^2,6,1,1/2^6}, {7^8,7^7,7^6,7^5,7^4,7^3,7^2,7,1,1/2^7}, {8^8,8^7,8^6,8^5,8^4,8^3,8^2,8,1,1/2^8}} the wolf cannot accept that many characters into their input box
 one year ago

amistre64 Group TitleBest ResponseYou've already chosen the best response.0
but f(0) would have amounted to the top right value
 one year ago

amistre64 Group TitleBest ResponseYou've already chosen the best response.0
1/10321920 if i dint mistype it up
 one year ago

amistre64 Group TitleBest ResponseYou've already chosen the best response.0
or 1 lets go with 1
 one year ago

amistre64 Group TitleBest ResponseYou've already chosen the best response.0
you are essentially matching a P8(x) to 2^(x) and at that many point so close together, we would expect it to be close enough to 2^0
 one year ago

amistre64 Group TitleBest ResponseYou've already chosen the best response.0
and ... now that i read it again .... f(0) is a point given in the interval that is set to 1/2^0 to begin with .... i need my mtDew :)
 one year ago

mukushla Group TitleBest ResponseYou've already chosen the best response.0
u actually set up a system of equation, 9 equations with 9 variables...still stuck, how u got f(0) ??
 one year ago

cruffo Group TitleBest ResponseYou've already chosen the best response.0
i have the same question since 2^i cannot equal 0 . I was thinkinf you would need the solve the system to find all tge coefficients of the polynomial.
 one year ago

cruffo Group TitleBest ResponseYou've already chosen the best response.0
is this close to what you are doing?
 one year ago

mukushla Group TitleBest ResponseYou've already chosen the best response.0
thats right
 one year ago

mukushla Group TitleBest ResponseYou've already chosen the best response.0
and yes close to what amistre did :)
 one year ago

cruffo Group TitleBest ResponseYou've already chosen the best response.0
prob is those values are large, 256^8 = 2^64.... there has to be a way other than brute force.
 one year ago

mukushla Group TitleBest ResponseYou've already chosen the best response.0
there must be a neater way...but i cant see a clue that will lead us to a nice solution
 one year ago

cruffo Group TitleBest ResponseYou've already chosen the best response.0
they are only asking for the constant, f(0) = a_0
 one year ago

cruffo Group TitleBest ResponseYou've already chosen the best response.0
is this from a particular class or book?
 one year ago

amistre64 Group TitleBest ResponseYou've already chosen the best response.0
lol, i see i used "i" instead of "2^i" for my point set
 one year ago

mukushla Group TitleBest ResponseYou've already chosen the best response.0
finally, key is defining \[g(x)=xf(x)1\]
 one year ago

amistre64 Group TitleBest ResponseYou've already chosen the best response.0
do explain
 one year ago

mukushla Group TitleBest ResponseYou've already chosen the best response.0
\(g(x)\) has the roots \(1,2,2^2,...2^8\) so we can write\[g(x)=a \ (x1)(x2)(x2^2)...(x2^8)\]
 one year ago

mukushla Group TitleBest ResponseYou've already chosen the best response.0
setting \(x=0\) we have\[g(0)=0\times f(0)1=1\]on the other hand\[1=g(0)=a (1)(2)(2^2)...(2^8)\]\[a=2^{(1+2+...+8)}=2^{36}\]
 one year ago

mukushla Group TitleBest ResponseYou've already chosen the best response.0
\(f(x)\) becomes\[f(x)=\frac{2^{36} \ (x1)(x2)(x2^2)...(x2^8)+1}{x}\]
 one year ago

mukushla Group TitleBest ResponseYou've already chosen the best response.0
then num of last expression has 0 as a root so the coefficient of \(x\) in num will be our answer
 one year ago

amistre64 Group TitleBest ResponseYou've already chosen the best response.0
ill have to review that later when ive got the time to better digest it, but good job nonetheless
 one year ago

mukushla Group TitleBest ResponseYou've already chosen the best response.0
ok, for checking, final answer is\[2\frac{1}{2^8}=\frac{511}{256}\]
 one year ago
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