Polynomial - Elementary Properties of Polynomials

Elementary Properties of Polynomials

• A sum of polynomials is a polynomial.
• A product of polynomials is a polynomial.
• A composition of two polynomials is a polynomial, which is obtained by substituting a variable of the first polynomial by the second polynomial.
• The derivative of the polynomial a_nxn + a_n-1xn-1 + ... + a₂x2 + a₁x + a₀ is the polynomial na_nxn-1 + (n-1)a_n-1xn-2 + ... + 2a₂x + a₁. If the set of the coefficients does not contain the integers (for example if the coefficients are integers modulo some prime number p), then ka_k should be interpreted as the sum of a_k with itself, k times. For example, over the integers modulo p, the derivative of the polynomial xp+1 is the polynomial 0.
• If the division by integers is allowed in the set of coefficients, a primitive or antiderivative of the polynomial a_nxn + a_n-1xn-1 + ... + a₂x2 + a₁x + a₀ is a_nxn+1/(n+1) + a_n-1xn/n + ... + a₂x3/3 + a₁x2/2 + a₀x +c, where c is an arbitrary constant. Thus x2+1 is a polynomial with integer coefficients whose primitives are not polynomials over the integers. If this polynomial is viewed as a polynomial over the integers modulo 3 it has no primitive at all.

Polynomials serve to approximate other functions, such as sine, cosine, and exponential.

All polynomials have an expanded form, in which the distributive and associative laws have been used to remove all brackets and commutative law has been used to make the like terms adjacent and combine them. All polynomials with coefficients in a unique factorization domain (for example, the integers or a field) also have a factored form in which the polynomial is written as a product of irreducible polynomials and a constant. In the case of the field of complex numbers, the irreducible polynomials are linear. For example, the factored form of

is

over the integers and

over the complex numbers.

Every polynomial in one variable is equivalent to a polynomial with the form

This form is sometimes taken as the definition of a polynomial in one variable.

Evaluation of a polynomial consists of assigning a number to each variable and carrying out the indicated multiplications and additions. Actual evaluation is usually more efficient using the Horner scheme:

In elementary algebra, methods are given for solving all first degree and second degree polynomial equations in one variable. In the case of polynomial equations, the variable is often called an unknown. The number of solutions may not exceed the degree, and equals the degree when multiplicity of solutions and complex number solutions are counted. This fact is called the fundamental theorem of algebra.

A system of polynomial equations is a set of equations in which each variable must take on the same value everywhere it appears in any of the equations. Systems of equations are usually grouped with a single open brace on the left. In elementary algebra, in particular in linear algebra, methods are given for solving a system of linear equations in several unknowns. If there are more unknowns than equations, the system is called underdetermined. If there are more equations than unknowns, the system is called overdetermined. Overdetermined systems are common in practical applications. For example, one U.S. mapping survey used computers to solve 2.5 million equations in 400,000 unknowns.

Viète's formulas relate the coefficients of a polynomial to symmetric polynomial functions of its roots.