Polynomials

Definition, Operations, Roots, and Applications

Basic (undergraduate years 1-2)

1. Definition of a Polynomial

Polynomial
A polynomial in the variable \( x \) is an expression of the form \[ p(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \] where \( a_0, a_1, \ldots, a_n \) are coefficients (constants) and \( n \) is a non-negative integer.

Basic terminology for polynomials:

Example
\[ p(x) = 3x^4 - 2x^3 + 5x - 7 \]
  • Degree: 4
  • Leading coefficient: 3
  • Constant term: -7

2. Types of Polynomials

2.1 Classification by Degree

Degree Name General Form Example
0 Constant polynomial \( a_0 \) \( 5 \)
1 Linear (degree-1) \( a_1 x + a_0 \) \( 2x + 3 \)
2 Quadratic (degree-2) \( a_2 x^2 + a_1 x + a_0 \) \( x^2 - 4x + 3 \)
3 Cubic (degree-3) \( a_3 x^3 + a_2 x^2 + a_1 x + a_0 \) \( x^3 + 2x^2 - 5 \)
\( n \) Degree-\( n \) polynomial \( \sum_{i=0}^{n} a_i x^i \) —

3. Polynomial Operations

3.1 Addition and Subtraction

Add or subtract coefficients of like-degree terms:

\[ (a_n x^n + \cdots + a_0) + (b_n x^n + \cdots + b_0) = (a_n + b_n) x^n + \cdots + (a_0 + b_0) \]
Example
\[ (3x^2 + 2x + 1) + (x^2 - 5x + 4) = 4x^2 - 3x + 5 \]

3.2 Multiplication

Multiply each pair of terms by distributivity:

\[ (a_m x^m + \cdots)(b_n x^n + \cdots) = \sum_{i,j} a_i b_j x^{i+j} \]
Example
\[ (x + 2)(x - 3) = x^2 - 3x + 2x - 6 = x^2 - x - 6 \]
📌 Degree of a product
\[ \deg(p \cdot q) = \deg(p) + \deg(q) \] (provided neither polynomial is zero)

3.3 Division

Division Algorithm
For polynomials \( f(x), g(x) \) with \( g \neq 0 \), there exist unique \( q(x), r(x) \) such that \[ f(x) = q(x) g(x) + r(x) \] where \( \deg(r) < \deg(g) \) or \( r = 0 \).
  • \( q(x) \): quotient
  • \( r(x) \): remainder
Proof.

We argue by induction on \( n = \deg(f) \). Let \( m = \deg(g) \).

Existence.

If \( n < m \), set \( q = 0,\; r = f \).

If \( n \geq m \), write \( f(x) = a_n x^n + \cdots \) and \( g(x) = b_m x^m + \cdots \) and set

\[ f_1(x) = f(x) - \frac{a_n}{b_m}\,x^{n-m}\,g(x). \]

The \( x^n \) term cancels, so \( \deg(f_1) < n \). By the inductive hypothesis, \( f_1 = g\,q_1 + r_1 \) with \( \deg(r_1) < m \), hence

\[ f = \frac{a_n}{b_m}\,x^{n-m}\,g + f_1 = g\!\left(\frac{a_n}{b_m}\,x^{n-m} + q_1\right) + r_1 \]

which is the required decomposition.

Uniqueness.

Suppose \( f = g\,q_1 + r_1 = g\,q_2 + r_2 \). Then \( g(q_1 - q_2) = r_2 - r_1 \). If \( q_1 \neq q_2 \), the left side has degree \( \geq m \) but the right side has degree \( < m \), a contradiction. Hence \( q_1 = q_2 \) and \( r_1 = r_2 \). \( \square \)

📌 Correspondence with long division
The step "subtract \( \frac{a_n}{b_m}\,x^{n-m}\,g(x) \) from \( f(x) \) to cancel the leading term" in the existence proof is exactly one step of polynomial long division. The induction on degree corresponds to repeating long-division steps.
Example: synthetic division
Divide \( x^3 - 2x^2 + x - 3 \) by \( x - 2 \): \[ x^3 - 2x^2 + x - 3 = (x - 2)(x^2 + 1) + (-1) \] Quotient: \( x^2 + 1 \); remainder: \( -1 \).

4. Roots of a Polynomial

Root / Zero
A value \( \alpha \) satisfying \( p(\alpha) = 0 \) is called a root or zero of the polynomial \( p(x) \).

4.1 Factor Theorem

Factor Theorem
\( \alpha \) is a root of \( p(x) \) ⇔ \( (x - \alpha) \) is a factor of \( p(x) \): \[ p(\alpha) = 0 \iff p(x) = (x - \alpha) q(x). \]
Proof.

By the division algorithm (§3.3), dividing \( p(x) \) by \( (x - \alpha) \) yields

\[ p(x) = (x - \alpha)\,q(x) + r \]

where \( \deg(r) < \deg(x - \alpha) = 1 \), so \( r \) is a constant.

Substituting \( x = \alpha \) gives

\[ p(\alpha) = (\alpha - \alpha)\,q(\alpha) + r = r. \]

Hence \( p(\alpha) = 0 \iff r = 0 \iff p(x) = (x - \alpha)\,q(x) \). \( \square \)

4.2 Fundamental Theorem of Algebra

Fundamental Theorem of Algebra
Every complex-coefficient polynomial of degree \( n \) has exactly \( n \) complex roots (counted with multiplicity).

Hence every degree-\( n \) polynomial factors as

\[ p(x) = a_n (x - \alpha_1)(x - \alpha_2) \cdots (x - \alpha_n). \]
📌 The "algebra" theorem that cannot be proved by algebra alone

Despite its name, no purely algebraic proof of this theorem is known. All known proofs use analytic ideas (Liouville's theorem in complex analysis) or topological arguments (degree of a continuous map). Gauss gave the first proof attempt in his 1799 doctoral dissertation, although it implicitly assumed the Jordan curve theorem. Gauss produced four different proofs in his lifetime.

The shortest proof uses Liouville's theorem (bounded entire functions are constant) and fits in a few lines. → Complex Analysis (Basic) Chapter 5: Proof of the Fundamental Theorem of Algebra

4.3 Elementary Symmetric Polynomials

Given \( n \) variables \( \alpha_1, \alpha_2, \ldots, \alpha_n \), the sum over all ways of choosing \( k \) of them and multiplying them together is called the \( k \)-th elementary symmetric polynomial.

Elementary Symmetric Polynomial
For \( n \) variables \( \alpha_1, \ldots, \alpha_n \), the \( k \)-th elementary symmetric polynomial \( e_k \) is defined by \[ e_k = e_k(\alpha_1, \ldots, \alpha_n) = \sum_{1 \leq i_1 < i_2 < \cdots < i_k \leq n} \alpha_{i_1}\,\alpha_{i_2}\cdots\alpha_{i_k} \] with \( e_0 = 1 \) (the empty product). For \( k > n \), \( e_k = 0 \).

Since the sum is over all ways of choosing \( k \) elements, the number of summands is \( \binom{n}{k} \).

Example: \( n = 2 \) (variables \( \alpha, \beta \))
\[ \begin{align} e_1 &= \alpha + \beta & &\text{(choose 1: sum)} \\ e_2 &= \alpha\beta & &\text{(choose 2: product)} \end{align} \]
Example: \( n = 3 \) (variables \( \alpha, \beta, \gamma \))
\[ \begin{align} e_1 &= \alpha + \beta + \gamma & &\text{(choose 1: }\binom{3}{1} = 3\text{ terms)} \\ e_2 &= \alpha\beta + \beta\gamma + \gamma\alpha & &\text{(choose 2: }\binom{3}{2} = 3\text{ terms)} \\ e_3 &= \alpha\beta\gamma & &\text{(choose 3: }\binom{3}{3} = 1\text{ term)} \end{align} \]

The polynomials are called "symmetric" because the value is invariant under any permutation of the variables. For example, in \( e_2(\alpha, \beta, \gamma) = \alpha\beta + \beta\gamma + \gamma\alpha \), swapping \( \alpha \) and \( \gamma \) yields \( \gamma\beta + \beta\alpha + \alpha\gamma \), the same value.

4.4 Relations between Roots and Coefficients (Vieta's Formulas)

Vieta's Formulas
For a degree-\( n \) polynomial \( p(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_0 \) with roots \( \alpha_1, \ldots, \alpha_n \), \[ \begin{align} \alpha_1 + \alpha_2 + \cdots + \alpha_n &= -\frac{a_{n-1}}{a_n} \\ \alpha_1 \alpha_2 + \alpha_1 \alpha_3 + \cdots &= \frac{a_{n-2}}{a_n} \\ \vdots \\ \alpha_1 \alpha_2 \cdots \alpha_n &= (-1)^n \frac{a_0}{a_n} \end{align} \]
Proof.

Step 1. The case \( a_n = 1 \).

By the fundamental theorem of algebra (Theorem 4.2), a monic degree-\( n \) polynomial factors as

\[ p(x) = (x - \alpha_1)(x - \alpha_2) \cdots (x - \alpha_n). \]

Expanding the product, each factor \( (x - \alpha_i) \) contributes either \( x \) or \( -\alpha_i \); summing over all choices gives the expansion.

The term \( x^{n-k} \) arises by choosing \( x \) from \( n-k \) factors and \( -\alpha_i \) from the remaining \( k \). The \( k \) choices correspond to a subset of indices \( \{i_1, \ldots, i_k\} \subseteq \{1, 2, \ldots, n\} \). One such choice contributes

\[ (-\alpha_{i_1})(-\alpha_{i_2})\cdots(-\alpha_{i_k}) = (-1)^k \,\alpha_{i_1}\alpha_{i_2}\cdots\alpha_{i_k}. \]

Summing over all subsets, the coefficient of \( x^{n-k} \) is

\[ (-1)^k \!\!\!\sum_{1 \leq i_1 < i_2 < \cdots < i_k \leq n} \!\!\!\alpha_{i_1}\alpha_{i_2}\cdots\alpha_{i_k} = (-1)^k \, e_k \]

where \( e_k \) is the \( k \)-th elementary symmetric polynomial (§4.3). Hence

\[ (x - \alpha_1)\cdots(x - \alpha_n) = \sum_{k=0}^{n} (-1)^k \, e_k \, x^{n-k}. \]

Writing this out,

\[ \begin{align} = x^n &- \underbrace{(\alpha_1 + \alpha_2 + \cdots + \alpha_n)}_{e_1}\,x^{n-1} \\ &+ \underbrace{(\alpha_1\alpha_2 + \alpha_1\alpha_3 + \cdots + \alpha_{n-1}\alpha_n)}_{e_2}\,x^{n-2} \\ &- \underbrace{(\alpha_1\alpha_2\alpha_3 + \cdots)}_{e_3}\,x^{n-3} + \cdots + (-1)^n\underbrace{\alpha_1\alpha_2\cdots\alpha_n}_{e_n}. \end{align} \]

On the other hand, the monic polynomial reads \( p(x) = x^n + a_{n-1}x^{n-1} + \cdots + a_1 x + a_0 \) with \( a_n = 1 \). Comparing the coefficient of \( x^{n-k} \),

\[ a_{n-k} = (-1)^k \, e_k \qquad (k = 1, 2, \ldots, n), \]

or equivalently

\[ e_k = (-1)^k \, a_{n-k} \qquad (k = 1, 2, \ldots, n). \]

This is Vieta's formula for monic polynomials.

Step 2. The general case (\( a_n \neq 1 \)).

Dividing \( p(x) = a_n x^n + a_{n-1}x^{n-1} + \cdots + a_0 \) by \( a_n \) yields

\[ \frac{p(x)}{a_n} = x^n + \frac{a_{n-1}}{a_n}x^{n-1} + \cdots + \frac{a_0}{a_n}, \]

a monic polynomial with the same roots \( \alpha_1, \ldots, \alpha_n \). The coefficient of \( x^{n-k} \) is \( a_{n-k}/a_n \), so applying Step 1 gives

\[ e_k(\alpha_1, \ldots, \alpha_n) = (-1)^k \,\frac{a_{n-k}}{a_n} \qquad (k = 1, 2, \ldots, n). \quad \square \]
Example: quadratic equation
For \( x^2 + px + q = 0 \) (so \( a_2 = 1,\; a_1 = p,\; a_0 = q \)) with roots \( \alpha, \beta \): \[ \alpha + \beta = -\frac{a_1}{a_2} = -p, \quad \alpha \beta = \frac{a_0}{a_2} = q. \]
Example: cubic equation
For \( x^3 + bx^2 + cx + d = 0 \) with roots \( \alpha, \beta, \gamma \): \[ \begin{align} \alpha + \beta + \gamma &= -b \\ \alpha\beta + \beta\gamma + \gamma\alpha &= c \\ \alpha\beta\gamma &= -d \end{align} \]

5. Special Polynomials

Monic Polynomial
A polynomial whose leading coefficient is 1: \[ p(x) = x^n + a_{n-1} x^{n-1} + \cdots + a_0. \]
Zero Polynomial
The polynomial with all coefficients zero. Its degree is undefined (or set to \( -\infty \)).

5.1 Chebyshev Polynomials

Chebyshev Polynomials
The Chebyshev polynomials of the first kind \( T_n(x) \) are defined by \[ T_n(\cos \theta) = \cos(n\theta) \] or, equivalently, by the recurrence \[ T_0(x) = 1, \quad T_1(x) = x, \quad T_{n+1}(x) = 2x T_n(x) - T_{n-1}(x). \]
Proof that the recurrence satisfies \( T_n(\cos\theta) = \cos(n\theta) \).

By induction on \( n \).

For \( n = 0 \): \( T_0(\cos\theta) = 1 = \cos 0 \). For \( n = 1 \): \( T_1(\cos\theta) = \cos\theta \). Both hold.

Assume the claim holds for \( n-1 \) and \( n \). The product-to-sum identity gives

\[ \cos((n+1)\theta) + \cos((n-1)\theta) = 2\cos\theta\,\cos(n\theta), \]

so

\[ \cos((n+1)\theta) = 2\cos\theta\,\cos(n\theta) - \cos((n-1)\theta). \]

Substituting the inductive hypotheses \( T_n(\cos\theta) = \cos(n\theta) \) and \( T_{n-1}(\cos\theta) = \cos((n-1)\theta) \),

\[ \cos((n+1)\theta) = 2\cos\theta\,T_n(\cos\theta) - T_{n-1}(\cos\theta) = T_{n+1}(\cos\theta). \]

Hence \( T_{n+1}(\cos\theta) = \cos((n+1)\theta) \). \( \square \)

5.2 Hermite Polynomials

Hermite Polynomials
\[ H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2}. \] They appear in the quantum harmonic oscillator.

5.3 Legendre Polynomials

Legendre Polynomials
\[ P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n} (x^2 - 1)^n. \] They form the basis of spherical harmonics.

6. Polynomial Rings

Polynomial Ring
The set \( R[x] \) of all polynomials in \( x \) over a ring \( R \) is itself a ring under polynomial addition and multiplication.

Properties of polynomial rings:

6.1 Irreducible Polynomials

Irreducible Polynomial
A polynomial \( p(x) \) over a field \( F \) is irreducible if it has no nontrivial factorization in \( F[x] \).
Examples
  • \( x^2 + 1 \) is irreducible over \( \mathbb{R} \) but reducible over \( \mathbb{C} \): \( (x-i)(x+i) \).
  • \( x^2 + x + 1 \) is irreducible over \( \mathbb{F}_2 \) (the field of characteristic 2).

7. Multivariate Polynomials

Multivariate Polynomial
A polynomial in several variables \( x_1, \ldots, x_n \): \[ p(x_1, \ldots, x_n) = \sum_{i_1, \ldots, i_n} a_{i_1 \cdots i_n} x_1^{i_1} \cdots x_n^{i_n}. \]
Total Degree
The total degree of the monomial \( x_1^{i_1} \cdots x_n^{i_n} \) is \( i_1 + \cdots + i_n \). The total degree of a polynomial is the maximum total degree of its monomials.
Example
\[ p(x, y) = x^3 y + 2xy^2 - 3x + 5 \] Total degree: 4 (from the monomial \( x^3 y \)).

8. Polynomial Interpolation

Lagrange Interpolation
Given \( n+1 \) points \( (x_0, y_0), \ldots, (x_n, y_n) \) with distinct \( x_i \), there exists a unique polynomial of degree \( \leq n \) passing through all of them: \[ p(x) = \sum_{i=0}^{n} y_i \prod_{j \neq i} \frac{x - x_j}{x_i - x_j}. \]
Proof.

Existence. For each \( i = 0, 1, \ldots, n \), define the Lagrange basis polynomial

\[ L_i(x) = \prod_{j \neq i} \frac{x - x_j}{x_i - x_j}. \]

This is a degree-\( n \) polynomial satisfying

\[ L_i(x_k) = \begin{cases} 1 & (k = i) \\ 0 & (k \neq i) \end{cases} \]

because when \( k \neq i \) the numerator contains the factor \( (x_k - x_k) = 0 \), and when \( k = i \) the numerator and denominator coincide.

Setting \( p(x) = \displaystyle\sum_{i=0}^{n} y_i\,L_i(x) \), we obtain a polynomial of degree \( \leq n \) with

\[ p(x_k) = \sum_{i=0}^{n} y_i\,L_i(x_k) = y_k, \]

passing through every given point.

Uniqueness. Suppose \( p(x) \) and \( q(x) \), both of degree \( \leq n \), pass through all given points. Their difference \( d(x) = p(x) - q(x) \) is a degree-\( \leq n \) polynomial with \( d(x_0) = \cdots = d(x_n) = 0 \). A polynomial of degree \( \leq n \) with \( n+1 \) roots must be identically zero, so \( d(x) = 0 \) and \( p = q \). \( \square \)

9. Applications

9.1 Numerical Computation

9.2 Coding Theory

9.3 Cryptography

9.4 Control Theory

10. References

11. Summary

Polynomials:

🔍 Related topics
Factorization, root formulas, the fundamental theorem of algebra, polynomial rings, irreducible polynomials, Vieta's formulas, Lagrange interpolation.