Glossary

1.3 Roots and factorization of polynomials

4 min read•july 30, 2024

Roots and factorization are key to understanding polynomials over fields. They help us break down complex equations into simpler parts, revealing their structure and solutions. This knowledge is crucial for solving polynomial equations and exploring field extensions.

Unique factorization of polynomials mirrors prime factorization of integers. It allows us to express polynomials as products of , providing insights into their properties and roots. This concept forms the foundation for more advanced topics in field theory.

Polynomial Roots over Fields

Finding Roots and Their Multiplicities

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  • A or p(x)p(x) is a value rr such that p(r)=0p(r) = 0
  • The states that every non-constant polynomial with complex coefficients has at least one
    • For example, the polynomial x2+1x^2 + 1 has no real roots, but it has the complex roots ii and −i-i
  • Over the real numbers, a polynomial of odd degree always has at least one , while a polynomial of even degree may have no real roots
    • The polynomial x3−1x^3 - 1 has the real root 11 and the complex roots −12±32i-\frac{1}{2} \pm \frac{\sqrt{3}}{2}i
  • The is the number of times it appears as a factor in the polynomial
    • A root with multiplicity 1 is called a
    • For instance, in the polynomial (x−1)2(x+1)(x - 1)^2(x + 1), the root 11 has multiplicity 2, and the root −1-1 has multiplicity 1

Properties and Techniques for Finding Roots

  • The sum of the multiplicities of all roots of a polynomial is equal to the degree of the polynomial
    • A polynomial of degree nn has exactly nn roots, counting multiplicities
  • can be used to determine the possible number of positive and negative real roots of a polynomial
    • The number of positive real roots is either equal to the number of sign changes between consecutive nonzero coefficients or is less than it by an even number
    • The number of negative real roots is the number of sign changes of f(−x)f(-x) or is less than it by an even number

Factoring Polynomials over Fields

Factoring over Different Fields

  • Factoring a polynomial involves expressing it as a product of irreducible polynomials, which cannot be factored further
  • Over the real numbers, a polynomial can be factored into a product of linear factors (corresponding to real roots) and irreducible quadratic factors (corresponding to pairs of roots)
    • The polynomial x3−1x^3 - 1 can be factored as (x−1)(x2+x+1)(x - 1)(x^2 + x + 1) over the real numbers
  • Over the complex numbers, the fundamental theorem of algebra guarantees that every polynomial can be factored into a product of linear factors
    • The polynomial x3−1x^3 - 1 can be factored as (x−1)(x−ω)(x−ω2)(x - 1)(x - \omega)(x - \omega^2) over the complex numbers, where ω=−12+32i\omega = -\frac{1}{2} + \frac{\sqrt{3}}{2}i is a cube root of unity
  • Over finite fields, polynomials can be factored using techniques such as the Berlekamp algorithm or Cantor-Zassenhaus algorithm

Irreducibility Criteria

  • can be used to determine the irreducibility of a polynomial with integer coefficients over the rational numbers
    • If a polynomial f(x)=anxn+an−1xn−1+⋯+a1x+a0f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 with integer coefficients satisfies:
      • pp divides each aia_i for i=0,1,…,n−1i = 0, 1, \ldots, n-1,
      • pp does not divide ana_n, and
      • p2p^2 does not divide a0a_0,
    • then f(x)f(x) is irreducible over the rational numbers
    • For example, the polynomial x3+3x+3x^3 + 3x + 3 is irreducible over Q\mathbb{Q} by Eisenstein's criterion with p=3p = 3

Euclidean Algorithm for Polynomials

The Euclidean Algorithm and Its Applications

  • The is an efficient method for finding the of two polynomials
  • The algorithm involves repeatedly dividing the polynomials and replacing the divisor with the remainder until the remainder is zero
    • The last non-zero remainder is the GCD
  • The can be used to find the coefficients of a linear combination of the polynomials that equals their GCD (Bézout's identity)
    • For polynomials f(x)f(x) and g(x)g(x), the extended Euclidean algorithm finds polynomials s(x)s(x) and t(x)t(x) such that s(x)f(x)+t(x)g(x)=gcd⁡(f(x),g(x))s(x)f(x) + t(x)g(x) = \gcd(f(x), g(x))

Properties of the GCD

  • The GCD of two polynomials is unique up to multiplication by a non-zero constant
  • If the GCD of two polynomials is 1, the polynomials are said to be relatively prime or coprime
    • For example, the polynomials x2+1x^2 + 1 and x3−1x^3 - 1 are coprime, as their GCD is 1

Unique Factorization of Polynomials

The Unique Factorization Theorem

  • The states that every non-zero polynomial over a field can be written as a product of irreducible polynomials in a unique way, up to the order of the factors and multiplication by non-zero constants
  • Irreducible polynomials over a field are the analogues of prime numbers in the ring of integers
    • An cannot be factored into a product of two polynomials of lower degree over the same field
    • For example, over the real numbers, the polynomial x2+1x^2 + 1 is irreducible, while over the complex numbers, it factors as (x+i)(x−i)(x + i)(x - i)

Applications and Implications

  • The unique factorization theorem allows for the development of a theory of divisibility for polynomials over fields, similar to the theory of divisibility for integers
    • For polynomials f(x)f(x) and g(x)g(x), we say f(x)f(x) divides g(x)g(x) if there exists a polynomial q(x)q(x) such that g(x)=q(x)f(x)g(x) = q(x)f(x)
  • The theorem is essential for understanding the structure of polynomial rings and their ideals, which play a crucial role in algebraic geometry and number theory
    • Polynomial rings over fields have many properties similar to the ring of integers, such as the existence of a division algorithm and the ability to perform polynomial long division

Key Terms to Review (22)

Complex conjugate: A complex conjugate of a complex number is formed by changing the sign of its imaginary part. For a complex number in the form $$a + bi$$, where $$a$$ and $$b$$ are real numbers and $$i$$ is the imaginary unit, the complex conjugate is $$a - bi$$. This concept is crucial when dealing with polynomials, as it helps in understanding the behavior of roots and their relationships during factorization.
Complex root: A complex root is a solution to a polynomial equation that can be expressed in the form of 'a + bi', where 'a' and 'b' are real numbers, and 'i' is the imaginary unit defined by 'i = \sqrt{-1}'. Complex roots often appear in conjugate pairs in polynomials with real coefficients, indicating that non-real solutions must always balance each other out to ensure the coefficients remain real.
Degree of a polynomial: The degree of a polynomial is the highest exponent of the variable in the polynomial expression. It provides crucial information about the behavior of the polynomial function, including its roots and factorization properties. Understanding the degree is essential for grasping concepts like minimal polynomials and algebraic degree, as well as how polynomials can be factored into linear and irreducible components.
Descartes' Rule of Signs: Descartes' Rule of Signs is a mathematical theorem that provides a way to determine the number of positive and negative roots of a polynomial based on the signs of its coefficients. This rule states that the number of positive real roots of a polynomial function is either equal to the number of sign changes between consecutive non-zero coefficients or less than it by an even integer, while for negative roots, you evaluate the polynomial at $-x$ and apply the same reasoning. This concept is essential for understanding how polynomials behave and how they can be factored effectively.
Eisenstein's Criterion: Eisenstein's Criterion is a powerful tool in algebra that provides a sufficient condition for determining the irreducibility of a polynomial with integer coefficients. Specifically, if a polynomial satisfies certain divisibility conditions with respect to a prime number, then it cannot be factored into lower-degree polynomials with integer coefficients. This criterion connects to the study of polynomial rings and the search for irreducible polynomials, as well as the analysis of roots and how polynomials can be factored over different fields.
Euclidean Algorithm: The Euclidean Algorithm is a method for computing the greatest common divisor (GCD) of two integers by repeatedly applying the principle that the GCD of two numbers also divides their difference. This technique is fundamental in number theory and has significant implications in polynomial factorization, as it can also be extended to find the GCD of polynomials, aiding in their roots and factorization.
Extended Euclidean Algorithm: The Extended Euclidean Algorithm is an extension of the Euclidean Algorithm, used for finding the greatest common divisor (GCD) of two integers, while also determining the coefficients that express this GCD as a linear combination of the two integers. This algorithm is vital for solving problems related to roots and factorization of polynomials, especially in number theory and algebra, since it provides a way to find multiplicative inverses in modular arithmetic, which is often necessary when working with polynomial equations.
Factoring Polynomials: Factoring polynomials is the process of breaking down a polynomial into simpler components, called factors, which when multiplied together yield the original polynomial. This concept is deeply connected to finding roots of polynomials, as each root corresponds to a factor of the polynomial, allowing for a clearer understanding of its behavior and solutions.
Fundamental theorem of algebra: The fundamental theorem of algebra states that every non-constant polynomial equation with complex coefficients has at least one complex root. This means that if you have a polynomial of degree $n$, it will have exactly $n$ roots in the complex number system, counting multiplicities. This theorem establishes a critical link between algebra and geometry, as it ensures that polynomials can be completely factored into linear factors over the complex numbers.
Greatest common divisor (gcd): The greatest common divisor (gcd) of two or more integers is the largest positive integer that divides each of the integers without leaving a remainder. This concept is crucial for understanding how to simplify fractions and factor polynomials, as it helps identify common factors that can be factored out, leading to clearer and simpler expressions.
Irreducible Factors: Irreducible factors are polynomials that cannot be factored into simpler polynomials over a given field or ring. This means they cannot be expressed as the product of lower-degree polynomials with coefficients in that same field or ring. Understanding irreducible factors is crucial when analyzing roots of polynomials, as these factors reveal the fundamental building blocks of polynomial equations.
Irreducible Polynomial: An irreducible polynomial is a non-constant polynomial that cannot be factored into the product of two non-constant polynomials over a given field. This concept is crucial in understanding the structure of field extensions, as the irreducibility of a polynomial often determines the nature of roots and their relationships within these fields.
Leading coefficient: The leading coefficient is the coefficient of the term with the highest degree in a polynomial. This important characteristic determines the polynomial's end behavior and influences its overall shape when graphed. The leading coefficient plays a significant role in understanding the properties of polynomials, particularly when discussing factorization and the irreducibility of polynomials.
Linear factor: A linear factor is a polynomial of degree one, which can be expressed in the form $ax + b$, where $a$ and $b$ are constants and $a \neq 0$. These factors represent the simplest building blocks of polynomials, and each linear factor corresponds to a unique root of the polynomial when set to zero. The connection between linear factors and the roots of polynomials is crucial for understanding polynomial factorization and solving polynomial equations.
Multiplicity of a root: Multiplicity of a root refers to the number of times a particular root appears in the factorization of a polynomial. If a polynomial can be factored as $(x - r)^m \cdot Q(x)$, where $Q(x)$ is another polynomial that does not have the root $r$, then we say that $r$ is a root of multiplicity $m$. This concept is important because it affects the behavior of the polynomial's graph at the root and plays a crucial role in understanding its overall structure.
Polynomial $p(x)$: A polynomial $p(x)$ is a mathematical expression that consists of variables, coefficients, and non-negative integer exponents, combined using addition, subtraction, and multiplication. Polynomials can be classified based on their degree, which is the highest power of the variable in the expression. The roots of a polynomial are the values of the variable that make the polynomial equal to zero, and understanding these roots is crucial for factorization and simplifying polynomials into products of linear factors.
Real root: A real root of a polynomial is a value of the variable for which the polynomial evaluates to zero, and this value is a real number. Real roots are significant because they indicate where the graph of the polynomial intersects the x-axis, and they play a crucial role in the factorization of polynomials, helping to express the polynomial in terms of its linear and irreducible factors.
Root: In mathematics, a root of a polynomial is a value for which the polynomial evaluates to zero. This concept is essential in understanding polynomial equations, as finding roots allows for the factorization of polynomials and insights into their structure, including the construction of splitting fields and the nature of separable polynomials.
Simple root: A simple root of a polynomial is a root that has multiplicity one, meaning it is a distinct solution to the polynomial equation where the factor corresponding to that root does not repeat. This concept is important because simple roots indicate points where the polynomial crosses the x-axis and contribute to the overall structure of the polynomial's factorization. Understanding simple roots helps in analyzing the behavior of polynomials and their minimal polynomials, as well as determining algebraic degrees.
Solution set: A solution set is the collection of all values that satisfy a given equation or system of equations. It plays a critical role in understanding how polynomials can be solved and factored, as the roots of a polynomial are precisely the values contained in its solution set. Identifying the solution set not only helps in finding the roots but also in determining the behavior and characteristics of the polynomial function itself.
Unique Factorization Theorem: The Unique Factorization Theorem states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the order of the factors. This theorem connects to polynomials, as it implies that polynomials can also be factored into irreducible elements, similar to how integers factor into primes. The uniqueness aspect emphasizes that this factorization holds true in terms of the polynomial's degree and coefficients, leading to essential applications in algebra and number theory.
Zero of a polynomial: A zero of a polynomial is a value for which the polynomial evaluates to zero. This means that if you substitute this value into the polynomial, the result will be zero, indicating that the polynomial intersects the x-axis at that point. Zeros are closely related to roots, as they represent solutions to the equation formed by setting the polynomial equal to zero, and they play a crucial role in polynomial factorization.
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