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

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

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

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

amistre64
 one year ago
Best ResponseYou've already chosen the best response.0rref{{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

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

amistre64
 one year ago
Best ResponseYou've already chosen the best response.01/10321920 if i dint mistype it up

amistre64
 one year ago
Best ResponseYou've already chosen the best response.0you 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

amistre64
 one year ago
Best ResponseYou've already chosen the best response.0and ... 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 :)

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

cruffo
 one year ago
Best ResponseYou've already chosen the best response.0i 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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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