Polynomial
You are a reviewer. Choose the entry on "polynomial" that will be accepted into the new edition of the Oxford Concise Dictionary of Mathematics.
(They are named with random greek letters for reference.)
IOTA
The term polynomial in x is an expression consisting of a variable x and coefficients from a field.
The degree of a polynomial is the highest power of x in the given polynomial.
The number of roots of a polynomial is no higher than the degree of the same polynomial.
A polynomial is reducible if it can be expressed as a product of polynomials with a smaller degree.
A polynomial is solvable by radicals if we can express its roots as a combination of its coefficients together with the basic operation +,-,*,/,^n,^(1/n) (where n is a natural number).
For the degree less than 5 there exists a formula for finding the roots, which can be expressed as a combination of basic operations.
KAPPA
- Definiton: A polynomial in x of a degree k is a sum of monomials (terms) the highest degree of which is k. Each monomial consists of a variable x raised to a certain power multiplied by a coefficient that is an element of R.
- Polynomial equations: a*x^k + b*x^(k-1)+…+z*x^1+c = 0
The solution of a polynomial equation consists of roots which could be described as values of x for which the polynomial is equal to 0. The polynomial is called reducible when there is an option to factorize the polynomial. - Any nth degree polynomial equation for n={1,2,3,4} is solvable by radicals and there exists an exact formula for each degree of the polynomial.
LAMBDA
A polynomial over a field is a finite sum of terms where a term consists of a coefficient and a variable raised to some power. The polynomial in x has the only variable x in each term. We define the degree of a polynomial as the highest power of x with a non-zero coefficient. A root of a univariate polynomial is a number which after being substituted for x in the polynomial gives zero. A polynomial is called reducible if it may be written as a product of two non-constant polynomials. Otherwise, a polynomial is called irreducible. A polynomial is solvable by radicals if each root can be given by a formula using constants, coefficients of the polynomial, addition, subtraction, multiplication, division and by taking n-th roots.