Polynomials

Suppose $w, z \in \mathbb {C}$. Verify the following equalities and inequalities.

Prove that if $w, z \in \mathbb {C}$, then $||w|-|z|| \leq |w-z|$. The inequality above is called the reverse triangle inequality.

Suppose $V$ is a complex vector space and $\varphi \in V^{\prime }$. Define $\sigma : V \rightarrow \mathbb {R}$ by $\sigma (v) = \mathrm{Re}\left(\varphi (v)\right)$ for each $v \in V$. Show that

for all $v \in V$.

Suppose $m$ is a positive integer. Is the set

a subspace of $\mathcal{P}(\mathbf{F})$?

Is the set

a subspace of $\mathcal{P}(\mathbf{F})$?

Suppose that $m$ and $n$ are positive integers with $m \leq n$, and suppose $\lambda_{1}, \ldots , \lambda_{m} \in \mathbf{F}$. Prove that there exists a polynomial $p \in \mathcal{P}(\mathbf{F})$ with $\operatorname {deg} p = n$ such that $0 = p\left(\lambda_{1}\right) = \cdots = p\left(\lambda_{m}\right)$ and such that $p$ has no other zeros.

Suppose that $m$ is a nonnegative integer, $z_{1}, \ldots , z_{m+1}$ are distinct elements of $\mathbf{F}$, and $w_{1}, \ldots , w_{m+1} \in \mathbf{F}$. Prove that there exists a unique polynomial $p \in \mathcal{P}_{m}(\mathbf{F})$ such that

for each $k = 1, \ldots , m+1$. This result can be proved without using linear algebra. However, try to find the clearer, shorter proof that uses some linear algebra.

Suppose $p \in \mathcal{P}(\mathbb {C})$ has degree $m$. Prove that $p$ has $m$ distinct zeros if and only if $p$ and its derivative $p^{\prime }$ have no zeros in common.

Prove that every polynomial of odd degree with real coefficients has a real zero.

For $p \in \mathcal{P}(\mathbb {R})$, define $T p: \mathbb {R} \rightarrow \mathbb {R}$ by

for each $x \in \mathbb {R}$. Show that $T p \in \mathcal{P}(\mathbb {R})$ for every polynomial $p \in \mathcal{P}(\mathbb {R})$ and also show that $T: \mathcal{P}(\mathbb {R}) \rightarrow \mathcal{P}(\mathbb {R})$ is a linear map.

Suppose $p \in \mathcal{P}(\mathbb {C})$. Define $q: \mathbb {C} \rightarrow \mathbb {C}$ by

Prove that $q$ is a polynomial with real coefficients.

Suppose $m$ is a nonnegative integer and $p \in \mathcal{P}_{m}(\mathbb {C})$ is such that there are distinct real numbers $x_{0}, x_{1}, \ldots , x_{m}$ with $p\left(x_{k}\right) \in \mathbb {R}$ for each $k = 0,1, \ldots , m$. Prove that all coefficients of $p$ are real.

Suppose $p \in \mathcal{P}(\mathbf{F})$ with $p \neq 0$. Let $U = \left\{ p q: q \in \mathcal{P}(\mathbf{F})\right\}$.

Suppose $p, q \in \mathcal{P}(\mathbb {C})$ are nonconstant polynomials with no zeros in common. Let $m = \operatorname {deg} p$ and $n = \operatorname {deg} q$. Use linear algebra as outlined below in (a)-(c) to prove that there exist $r \in \mathcal{P}_{n-1}(\mathbb {C})$ and $s \in \mathcal{P}_{m-1}(\mathbb {C})$ such that