Let n ∈ℤ+ be a positive integer, let w0, w1, …, wn ∈ℂ be distinct complex numbers, and let z0, z

step 1 Hi there, so for this problem we are told the fundamental theorem of algebra states that for a c

Let n ∈ℤ+ be a positive integer, let w0, w1, …, wn ∈ℂ be...

Let $n \in \mathbb{Z}_{+}$ be a positive integer, let $w_{0}, w_{1}, \ldots, w_{n} \in \mathbb{C}$ be distinct complex numbers, and let $z_{0}, z_{1}, \ldots, z_{n} \in \mathbb{C}$ be any complex numbers. Then one can prove that there is a unique polynomial $p(z)$ of degree at most $n$ such that, for each $k \in\{0,1, \ldots, n\}, p\left(w_{k}\right)=z_{k}$ (a) Find the unique polynomial of degree at most 2 that satisfies $p(0)=0, p(1)=1$, and $p(2)=2$. (b) Can your result in Part (a) be easily generalized to find the unique polynomial of degree at most $n$ satisfying $p(0)=0, p(1)=1, \ldots, p(n)=n ?$

Let $n \in \mathbb{Z}_{+}$ be a positive integer, let $w_{0}, w_{1}, \ldots, w_{n} \in \mathbb{C}$ be distinct complex numbers, and let $z_{0}, z_{1}, \ldots, z_{n} \in \mathbb{C}$ be any complex numbers. Then one can prove that there is a unique polynomial $p(z)$ of degree at most $n$ such that, for each $k \in\{0,1, \ldots, n\}, p\left(w_{k}\right)=z_{k}$
(a) Find the unique polynomial of degree at most 2 that satisfies $p(0)=0, p(1)=1$, and $p(2)=2$.
(b) Can your result in Part (a) be easily generalized to find the unique polynomial of degree at most $n$ satisfying $p(0)=0, p(1)=1, \ldots, p(n)=n ?$

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