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## Homework Statement

(This problem is from the Spivak 2nd Ed. I had to translate it from spanish since my book is in spanish)

If [tex]x_1, \ldots, x_n[/tex] are different numbers, find a polynomial function [tex]f_i[/tex] of [tex]n-1[/tex] degree that takes value 1 on [tex]x_i[/tex] and 0 in [tex]x_j[/tex] for [tex]j \neq i[/tex]. Indication: the product of every [tex](x-x_j)[/tex] for [tex]j \neq i[/tex] is 0 if [tex]j \neq i[/tex].

\prod_{j=1}^{n} (x-x_{j})

## Homework Equations

[tex]\prod_{j=1}^{n} (x-x_{j})[/tex]

## The Attempt at a Solution

So far... Well so basically I stated all the known and unknown but I can't seem to get

past that. So here's what I have...

There is a set of [tex]x_1, \ldots, x_n[/tex]

[tex]f_{i}[/tex] is of [tex]n-1[/tex] degree.

There's a function such

[tex]f_{i}(x) = a_{n-1}x^{n-1} + a_{n-2}x^{n-2} + \ldots + a_{1}x + a_{0}[/tex]

There's a pair [tex](x_{i}, f_{i}(x_{i})[/tex] such that

[tex]f_{i}(x_{i}) = a_{n-1}x_{i}^{n-1} + a_{n-2}x_{i}^{n-2} + \ldots + a_{1}x_{i} + a_{0} = 1[/tex]

And there's also a pair [tex](x_{j}, f_{i}(x_{j}))[/tex] such that

[tex]f_{i}(x_{j}) = a_{n-1}x_{j}^{n-1} + a_{n-2}x_{j}^{n-2} + \ldots + a_{1}x_{j} +a_{0} = 0[/tex]

But I can't seem to connect the indication with the whole problem... any help? Oh, and I posted it in calculus but I am not quiet sure if this belongs in precalculus forum instead. I am sorry if this doesn't belong here.

Thanks for any advice in advance.