Let P be a polynomial with complex coefficients: P(z):=an...

Let $P$ be a polynomial with complex coefficients: $$ P(z):=a_{n} z^{n}+a_{n-1} z^{n-1}+\cdots+a_{0} \text { with } n \in \mathbb{N}_{0}, a_{\nu} \in \mathbb{C}, \text { for } 0 \leq \nu \leq n $$ A real or complex number $\zeta$ is called a root or a zero of $P$, if $P(\zeta)=0$ Show: If all the coefficients $a_{\nu}$ are real, then we have $$ P(\zeta)=0 \Longrightarrow P(\bar{\zeta})=0 $$ In other words, if the polynomial $P$ has only real coefficients then the roots of $P$ which are not real occur as pairs of complex conjugate numbers.

Let $P$ be a polynomial with complex coefficients:
$$
P(z):=a_{n} z^{n}+a_{n-1} z^{n-1}+\cdots+a_{0} \text { with } n \in \mathbb{N}_{0}, a_{\nu} \in \mathbb{C}, \text { for } 0 \leq \nu \leq n
$$
A real or complex number $\zeta$ is called a root or a zero of $P$, if $P(\zeta)=0$
Show: If all the coefficients $a_{\nu}$ are real, then we have
$$
P(\zeta)=0 \Longrightarrow P(\bar{\zeta})=0
$$
In other words, if the polynomial $P$ has only real coefficients then the roots of $P$ which are not real occur as pairs of complex conjugate numbers.

Key Concepts

Conjugate Root Theorem

The conjugate root theorem states that for any polynomial with real coefficients, if a complex number is a root, then its complex conjugate is also a root. This occurs because the algebraic structure of the polynomial and the real nature of the coefficients ensure that taking the conjugate of the entire polynomial's evaluation yields the evaluation of the polynomial at the conjugate of the original root, thereby forcing the conjugate to also be a zero.

Polynomials with Real Coefficients

A polynomial with real coefficients is one in which all the coefficients are real numbers. This property is essential because it makes many aspects of the polynomial's behavior, such as its roots, subject to the reality constraint. In the context of complex numbers, when the coefficients are real, the evaluation of the polynomial at the conjugate of any complex number interacts with the polynomial's structure in a predictable way.

Complex Conjugation

Complex conjugation is an operation that takes a complex number and reflects it across the real axis. For a complex number z = a + bi, its conjugate is given by a - bi. This operation has important algebraic properties such as preserving the modulus and reversing the sign of the imaginary part, which is critical in showing how polynomial evaluations behave when applied to conjugate pairs.

Properties of Complex Conjugation in Algebra

The operation of complex conjugation preserves addition, subtraction, and multiplication. That is, conjugate of (z+w) is the sum of the conjugates, and similarly for products and powers. In the context of polynomial evaluation, this means that when each term a_n*z^n is conjugated, the result is exactly a_n*(conjugate of z)^n because the coefficients being real are unaffected by conjugation.

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