Definitions

1. polynomial ring
2. polynomial ring over integral domain
3. polynomial ring over a field
4. skew polynomial ring
5. skew polynomial ring
6. order and degree of polynomial
7. length of a polynomial
8. polynomial growth

Polynomials over

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. polynomial congruence

Roots of polynomials

1. fundamental theorem of algebra
2. continuous dependence on coefficients
3. Tchirnhaus Transformations
4. rational root theorem
5. polynomial equation of odd degree
6. Viete formulas
7. Descartes' rule of signs
8. Eisenstein criterion
9. homogeneous equation

Polynomials with special properties

1. zero polynomial
2. opposite polynomial
3. symmetric polynomial
4. all one polynomial
5. primitive polynomial
6. homogeneous polynomial
7. weighted homogeneous polynomial
8. monic polynomial
9. irreducible polynomial
10. irreducibility of binomials with unity coefficients
11. reciprocal polynomial
12. orthogonal polynomials

Classical orthogonal polynomials

1. Chebyshev polynomials
2. Hermite polynomials
3. Laguerre polynomials
4. Legendre polynomials
5. Jacobi polynomials

Other polynomial families

1. Bernoulli polynomials and numbers
2. Bernoulli polynomials
3. Bernstein polynomials
4. Lamé polynomials
5. Boubaker polynomials

Miscellaneous applications

1. Taylor polynomial
2. Characteristic polynomial in linear algebra (see also Cayley-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