Spivak's Calculus - Capital-Pi/Function Proof

[Fllorv]

Homework Statement

If x_1,...,x_n are distinct numbers, find a polynomial function f_i of degree n-1 which is $1$ at x_i and $0$ at x_j for j=/=i. (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

$n$
\prod(x-x_j)
j=1
j=/=i

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
f_i (x) =
$n$
\prod(x-x_j)
j=1
j=/=i
$divided$ $by$
$n$
\prod(x_i - x_j)
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 f_i
-- 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 x_j and x_i 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

 

[lurflurf]

That is nearly unintelligible, but I think you are talking about
http://mathworld.wolfram.com/LagrangeInterpolatingPolynomial.html

-- I'm not sure why the function needs to be subscripted as fi
You are not finding a function, but n functions hence the subscript
-- 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?
x is the variable we are finding functions, not numbers
-- What CAN xj and xi equal? Anything that equals 1 and 0 respectively (in the function)?
the xi are distinct numbers
-- What does it mean that j=1 , does it mean that it is the first number in the previously mentioned series?
x1 is the first number, though it does not say the numbers are in order, it would not matter.

Maybe you will understand the general case if you do one specific example
find 4 cubic polynomials f1,f2,f3,f4 such that
f1(0)=1
f1(0)=0
f1(0)=0
f1(0)=0
f2(1)=0
f2(1)=1
f2(1)=0
f2(1)=0
f3(3)=0
f3(3)=0
f3(3)=1
f3(3)=0
f4(4)=0
f4(4)=0
f4(4)=0
f4(4)=1

expressed more briefly as
f_i(x_j)=δ_ij
where i,j=1,2,3,4
(x1,x2,x3,x4)=(0,1,3,4)
and
δ_ij=1 when i=j
δ_ij=0 otherwise