1. ## Polynomial

Question: If $\displaystyle x_{1}, ... , x_{n}$ are distinct numbers, find a polynomial $\displaystyle f_{i}$ of degree $\displaystyle n-1$ which is $\displaystyle 1$ at $\displaystyle x_{i}$ and 0 at $\displaystyle x_{j}$ for $\displaystyle j \ne i$. Hint: the product of all $\displaystyle (x-x_{i})$ for $\displaystyle j \ne i$ is zero at $\displaystyle x_{j}$ if $\displaystyle j \ne i$. .

Confusion: The part of the question that I don't understand is the 'which is $\displaystyle 1$ at $\displaystyle x_{i}$ and 0 at $\displaystyle x_{j}$ for $\displaystyle j \ne i$.' What are $\displaystyle x_{i}$ and $\displaystyle x_{j}$? Does it mean that for any two numbers we choose from the sequence $\displaystyle x_{1}, ..., x_{n}$, we get $\displaystyle f_{i}$ as 0, for one, and 1, for the other? And how would you go about finding this polynomial?

2. The polynomial is...

$\displaystyle \displaystyle P(x)= \frac{\prod_{i=1, i\ne j}^{n} (x-x_{i})} {\prod_{i=1, i\ne j}^{n} (x_{j}-x_{i})}$

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$

3. Originally Posted by chisigma
The polynomial is...

$\displaystyle \displaystyle P(x)= \frac{\prod_{i=1, i\ne j}^{n} (x-x_{i})} {\prod_{i=1, i\ne j}^{n} (x_{j}-x_{i})}$

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$
Can you do a bit of explanation, please, if you don't mind? For example, it's not apparent to me that it's of degree $\displaystyle n-1$.

4. Originally Posted by Hardwork
The part of the question that I don't understand is the 'which is $\displaystyle 1$ at $\displaystyle x_{i}$ and 0 at $\displaystyle x_{j}$ for $\displaystyle j \ne i$.' What are $\displaystyle x_{i}$ and $\displaystyle x_{j}$?
The problem asks to find, for every $\displaystyle i=1,\dots,n$, a polynomial $\displaystyle f_i$ with the given property, which is a total of $\displaystyle n$ polynomials.

For example, it's not apparent to me that it's of degree $\displaystyle n - 1$.
In the denominator, you have a product of numbers. In the numerator, you have a product of $\displaystyle n-1$ expressions of the form $\displaystyle x-c$ for various constants $\displaystyle c$.

This is similar to the interpolation polynomial in the Lagrange form (the link uses notation $\displaystyle \ell_j$ instead of $\displaystyle f_j$).