Note 1
[A] Polynomials
1.1 Foundations
Before we get into the world of polynomials in olympiad algebra, let’s discuss some notations that we will be using throughout the notes and key theorems as a review. Please feel free to skip over to the next section if you are already familiar with the topics below.
Definition 1.1.1
A monomial is a single term composed with variables raised to powers of whole numbers and a coefficient. Polynomial, denoted as \(P(x)\), is a sum of finitely many monomials.
Some examples of monomial includes \(x, 3a^2, 9, (5-2i)x^3y^2\) while \(x^{-2}, a + b, \sin {x}\) are not monomials. Polynomials are typically written with the following sum notation. \[ P(x) = \sum _{i=0}^n a_i x^i = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \] With this in mind, we can define some notations that occasionally appear.
Definition 1.1.2
Below are common set notations for polynomials where \(n\) is a non-negative integer. \begin{align*} \mathbb {Z}[x] &\coloneqq \left \{ \sum _{i=0}^n a_i x^i \mid a_i \in \mathbb {Z} \right \} \\ \mathbb {Q}[x] &\coloneqq \left \{ \sum _{i=0}^n a_i x^i \mid a_i \in \mathbb {Q} \right \} \\ \mathbb {R}[x] &\coloneqq \left \{ \sum _{i=0}^n a_i x^i \mid a_i \in \mathbb {R} \right \} \\ \mathbb {C}[x] &\coloneqq \left \{ \sum _{i=0}^n a_i x^i \mid a_i \in \mathbb {C} \right \} \end{align*}
Continuing with the notations, let’s discuss some key theorems that cannot be omitted when discussing polynomials.
Theorem 1.1.3: Fundamental Theorem of Algebra
Any nonconstant polynomial \(P(x) \in \mathbb {C}[x]\) has at least one complex root [MATH01-3].
There are multiple proofs for this beautiful theorem, but I think they are out of scope for the notes. Trust me, the theorem is true, and maybe I will include a rigorous proof of the statement in other parts of LibreNotebook. Now this theorem can lead us to another very beautiful proven assumption that we can use when solving problems.
Corollary 1.1.4
All polynomials \(P(x) \in \mathbb {C}[x]\) with \(\deg (P) = n > 1\) can be factored into \(n\) linear factors with multiplicity.
Proof.
Notice that by FTA, \(P_n(x)\) can be represented as \(P_n(x) = a (x - z_1) P_{n-1}(x)\) where \(P_k(x)\) is a polynomial with degree \(k\). Continuing recursively until \(n = 1\), \(P_n(x)\) can be written as the following. \begin{align*} P_n(x) &= a (x - z_1) (x - z_2) \cdots (x - z_{n-1}) P_1 (x) \\ &= a (x - z_1) (x - z_2) \cdots (x - z_{n-1}) (x - z_n) \end{align*}
Therefore, any polynomial with degree \(n > 1\) can be factored into exactly \(n\) linear factors with multiplicity.
The corollary also implies that there exists \(n\) roots counting multiplicity. Another key property to remember is remainder theorem and factor theorem. They are pretty self-evident, so I will omit the proof here, but below is the simple statement.
Theorem 1.1.5
For any polynomial \(P\), if the polynomial is divided by \(x - a\), then the remainder is \(P(a)\). Moreover if \(P(a) = 0\), then \((x - a)\) is a factor of \(P\).
The theorem above is basically stating that polynomials can be written as the following. \[ P(x) = (x - a) Q(x) + P(a) \] This is pretty self evident as we get \(P(a) = P(a)\) if we substitute \(x = a\) and \(Q(x)\) is the quotient polynomial.
Before we finish our review, here is the final theorem.
Theorem 1.1.6: Rational Root Theorem
Every rational root \(\frac {p}{q}\) in lowest terms of a polynomial \(P(x) = \sum _{i=0}^n a_i x^i\) with \(a_0, a_n \neq 0\) satisfies the following conditions: \(p \mid a_0\) and \(q \mid a_n\).
Proof.
First and foremost, notice that if \(\frac {p}{q}\) is a solution to \(P(x) = 0\), then the following equations hold. \begin{align*} P \left ( \frac {p}{q} \right ) = \sum _{i=0}^n a_i \left ( \frac {p}{q} \right )^i &= 0 \\ a_0 q^n + a_1 p q^{n-1} + a_2 p^2 q^{n-2} + \cdots + a_n p^n &= 0 \end{align*}
Therefore, the following equations can be derived. \begin{align*} q \left ( a_0 q^{n-1} + a_1 p q^{n-2} + a_2 p^2 q^{n-3} + \cdots + a_{n-1} p^{n-1} \right ) &= -a_n p^n \\ p \left ( a_1 q^{n-1} + a_2 p q^{n-2} + a_3 p^2 q^{n-3} + \cdots + a_n p^{n - 1} \right ) &= -a_0 q^n \end{align*}
Because \(p\) and \(q\) are coprime, the first equation reveals that \(q \mid a_n\) and the second equation shows that \(p \mid a_0\).
This theorem sometimes comes handy if we had to blindly guess a solution to further factorize as it narrows down our choices. Now with the reminders, let’s discuss the topic that we all may have heard of: Roots of Unity.