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Mathematics LibreTexts

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 \(p(\bar{\alpha}) = 0.\)