Discrete Mathematics - Vol 3 | 21. Roots of a Polynomial by Abraham | Learn Smarter
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21. Roots of a Polynomial

The chapter explores the concept of roots of polynomials over a field, establishing that a polynomial of degree n can have at most n roots. It further discusses methods for finding irreducible factors of polynomials, particularly focusing on small degree polynomials in the context of integers. It introduces the concept of monic polynomials and provides a step-by-step method for checking linear and quadratic factors, culminating in the factorization of x^4 + 1.

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Sections

  • 21.

    Roots Of A Polynomial

    This section defines the roots of a polynomial and explores their properties, particularly within the context of the factor theorem.

    Learning Practice

  • 21.1

    Definition Of Roots

    The section explains the concept of roots of polynomials within a field, detailing their definitions, properties, and relevance to polynomial factorization.

    Learning Practice

  • 21.2

    Number Of Roots For Degree N Polynomial

    This section discusses the number of roots a polynomial of degree n can possess, establishing that it can have at most n roots based on the factor theorem.

    Learning Practice

  • 21.2.1

    Proof Of Roots Upper Bound

    This section discusses the roots of polynomials over a field, specifically how the degree of a polynomial determines the maximum number of roots it can have.

    Learning Practice

  • 21.2

    Finding Irreducible Factors

    This section details how to identify irreducible factors of polynomials using the factor theorem and explores examples of monic polynomials.

    Learning Practice

  • 21..1

    Methods For Finding Irreducible Factors

    The section outlines methods for finding irreducible factors of polynomials, emphasizing the importance of roots and the factor theorem.

    Learning Practice

  • 21..2

    Example Of A Polynomial Factorization

    This section discusses the concept of polynomial roots and how to determine the number of roots for a polynomial of degree n, as well as methods for factorizing polynomials.

    Learning Practice

  • 21.2.1

    Possibilities Of Factors

    This section explores the definition of polynomial roots, the maximum number of roots a polynomial can have, and methods for finding irreducible factors of polynomials.

    Learning Practice

  • 21.2.2

    Conditions For Quadratic Factors

    This section discusses the concept of roots in polynomials, focusing on the maximum number of roots a polynomial can have based on its degree and explores methods for finding irreducible factors.

    Learning Practice

  • 21.2.3

    Solving The Equations

    This section discusses the concept of roots of polynomials and the significance of the factor theorem in determining the number of roots.

    Learning Practice

References

ch68 - part B.pdf

Class Notes

Memorization

What we have learnt

  • Polynomials of degree n can...
  • Roots of a polynomial f(x) ...
  • Finding irreducible factors...

Final Test

Revision Tests