Question 6

a)

Find $f_{i}(x)n-1$ degree polynomial where: $f_{i}(x_{i})=1$ and $f_{i}(x_{j})=0$

Noting that: $f_{i}(x)=x^{n-1}+...$

is the same as $f_{i}(x)=\prod _{j=1}^{n}x_{j}+...$

Then:

$f_{i}(x)=\prod _{j=1}^{n}{\frac {(x-x_{j})}{(x_{i}-x_{j})}}$ where $j!=i$

Now find a polynomial function $f$ of degree $n-1$ such that $f(x_{i})=a_{i}$

$f(x)=\sum _{i=1}^{n}a_{i}f_{i}(x)$ where $j!=i$

so: $f(x)=\sum _{i=1}^{n}a_{i}\prod _{j=1}^{n}{\frac {(x-x_{j})}{(x_{i}-x_{j})}}$ where $j!=i$

(Note that in this equation will always resulting in 0 unless x = x_i)