topics on polynomials

Definitions

1. polynomial ring

2. derivative of polynomial

3. polynomial ring over integral domain

4. polynomial ring over a field

5. polynomial ring which is PID

6. skew polynomial ring

7. $q$ skew polynomial ring

8. factor theorem

9. order and degree of polynomial

10. length of a polynomial

11. polynomial growth

Polynomials over $ℝ$

1. polynomial long division

2. polynomial interpolation, Lagrange interpolation formula

3. Weierstrass approximation theorem

4. Sturm’s theorem

5. properties of quadratic equation

6. cubic formula

7. quartic formula

8. Abel’s theorem

9. algebraic equation

10. square root of polynomial

11. casus irreducibilis

12. conditional congruence

Roots of polynomials

1. zero of polynomial

2. fundamental theorem of algebra

3. continuous dependence on coefficients

4. Tchirnhaus transformations

5. rational root theorem

6. polynomial equation of odd degree

7. Viete formulas

8. Descartes’ rule of signs

9. Eisenstein criterion

10. zeros of polynomial derivative

11. homogeneous equation

Polynomials with special properties

1. zero polynomial

2. opposite polynomial

3. symmetric polynomial

4. monic polynomial

5. minimal polynomial

6. all one polynomial

7. prime factors of $x^n-1$ (http://planetmath.org/PrimeFactorsOfXn1)

8. primitive polynomial

9. homogeneous polynomial

10. weighted homogeneous polynomial

11. irreducible polynomial

12. irreducibility of binomials with unity coefficients

13. reciprocal polynomial

14. orthogonal polynomials

Classical orthogonal polynomials

1. Chebyshev polynomials

2. Hermite polynomials

3. Laguerre polynomials

4. Legendre polynomials

5. Jacobi polynomials

6. properties of orthogonal polynomials

Other polynomial families

1. Bernoulli polynomials and numbers

2. Bernoulli polynomials

3. Bernstein polynomials

4. http://planetmath.org/node/12200Boubaker polynomials

5. Euler polynomials

6. Lamé polynomials

Miscellaneous applications

1. Taylor polynomial

2. Characteristic polynomial in linear algebra (see alsoCayley-Hamilton theorem)

Related concepts

1. division algorithm for polynomials

2. since sets of polynomial are rings, commutative algebra concepts like PID, UFD and Euclidean domains are heavily related

3. Gröbner basis