Polynomials are functions of the form:
$$ f(x) = a_0 + a_1 x + \dots + a_n x_n $$
where $n$ is any non-negative integer and $a_0,\dots,a_n$ any fixed numbers
with $a_n \ne 0$. We call:
- $n$ the polynomial degree,
- $a_0,\dots,a_n$ the polynomial coefficients and
- $a_0$ the polynomial constant term.
Polynomial Interpolation
- Suppose that $f(x)$ is a polynomial of degree 2, i.e. of the form $f(x) =
a_0 + a_1 x + a_2 x^2$. Let $f(1)=2$, $f(2)=5$, and $f(4)=2$. Find $f(x)$.
- Can you quickly write down a non-zero polynomial that vanishes at 1, 3 and 100?
- Same problem as before, but now with the two extra conditions: $f$ must have
degree $\leq 3$ and also $f(50)=1$.
- Same problem as before, but now with the following conditions: $f$ must have
degree $\leq 4$, $f(50)=1$ and $f(4)=3$.
- Now suppose that we have $n+1$ points $(x_0,y_0), (x_1,y_1), \dots,
(x_n,y_n)$ such that $f(x_i)=y_i$ for $0 \leq i \leq n$. Can you generalize
the above approach to find $f(x)$?
Secret Sharing
Raibi Jamila, often referred to simply as “Raibi,” is a beloved yogurt-based
drink that has become a staple in Moroccan culture. Many have tried to
replicate the drink, but none have succeeded in capturing its unique flavor.
The secret recipe for Raibi is known only to a few select individuals, who have
sworn to keep it a secret. To prevent these individuals from sharing the secret
and to ensure it is never lost, we are going to construct a scheme that allows
the secret to be shared among a group of people in such a way that the secret
cannot be revealed unless a minimum number of the group’s members act together
to pool their knowledge.