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)