
# 3.E: Exercises for Chapter 3

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## 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

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