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Fllorv
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Homework Statement
If [itex]x_1,...,x_n[/itex] are distinct numbers, find a polynomial function [itex]f_i[/itex] of degree [itex]n-1[/itex] which is ##1## at [itex]x_i[/itex] and ##0## at [itex]x_j[/itex] for [itex]j=/=i.[/itex] (This product is usually denoted by [see below] the symbol Capital Pi playing the same role for products that Capital Sigma plays for sums.)
Homework Equations
##n##
[itex]\prod(x-x_j)[/itex]
[itex]j=1[/itex]
[itex]j=/=i[/itex]
The Attempt at a Solution
This is a question out of Spivak's Calculus (particularly Chapter Three, Question Six). I've looked in the answer book, and he comes to the conclusion that
[itex]f_i (x) =[/itex]
##n##
[itex]\prod(x-x_j)[/itex]
j=1
j=/=i
##divided## ##by##
##n##
[itex]\prod(x_i - x_j)[/itex]
j=1
j=/=i
But I'm not satisfied with this answer alone
I'm almost positive that I'm over-complicating this problem, so I need help simplifying it.
-- I'm not sure why the function needs to be subscripted as [itex]f_i[/itex]
-- I'm not sure where the un-subscripted ##x## comes from in the Pi equation; does it rely on anything, or it just any random number?
-- What CAN [itex]x_j[/itex] and [itex]x_i[/itex] equal? Anything that equals 1 and 0 respectively (in the function)?
-- What does it mean that ##j=1## , does it mean that it is the first number in the previously mentioned series?
I suppose I have more questions, but I've confused myself and really need help being walked through it.
Thanks
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