3.E: Exercises for Chapter 3

Calculational Exercises

1. 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 $k \in \{0, 1, . . . , n\}, p(w_k ) = z_k.$

(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 $p(0) = 0, p(1) = 1, \ldots , p(n) = n$?

2. Given any complex number $\alpha \in \mathbb{C},$ show that the coefficients of the polynomial

$$(z − \alpha)(z − \bar{\alpha})$$

are real numbers.

Proof-Writing Exercises

1. Let $m, n \in \mathbb{Z}_+$ be positive integers with $m \leq n$. Prove that there is a degree n polynomial $p(z)$ with complex coefficients such that $p(z)$ has exactly m distinct roots.

2. Given a polynomial $p(z) = a_n z^n + \cdots + a_1 z + a_0$ with complex coefficients, define the conjugate of $p(z)$ to be the new polynomial

$$\bar{p}(z) = \bar{a_n} z^n + \cdots + \bar{a_1}z + a_0.$$

(a) Prove that $\bar{p(z)} = \bar{p}(\bar{z}).$
(b) Prove that $p(z)$ has real coefficients if and only if $\bar{p}(z) = p(z).$
(c) Given polynomials $p(z), q(z),$ and $r(z)$ such that $p(z) = q(z)r(z),$ prove that $\bar{p}(z) = \bar{q}(z)\bar{r}(z).$

3. Let $p(z)$ be a polynomial with real coefficients, and let $\alpha \in \mathbb{C}$ be a complex number.
Prove that $p(\alpha) = 0$ if and only $p(\bar{\alpha}) = 0.$

Contributors

• Isaiah Lankham, Mathematics Department at UC Davis
• Bruno Nachtergaele, Mathematics Department at UC Davis
• Anne Schilling, Mathematics Department at UC Davis

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