Elementary properties of polynomials

1.A sum of polynomials is a polynomial
2.A product of polynomials is a polynomial
3.The derivative of a polynomial function is a polynomial function
4.Any primitive or antiderivative of a polynomial function is a polynomial function

Polynomials serve to approximate other functions, such as sine, cosine, and exponential.
All polynomials have an expanded form, in which the distributive law has been used to remove all parentheses. All polynomials with real or complex coefficients also have a factored form in which the polynomial is written as a product of linear polynomials. For example, the polynomial

is the expanded form of the polynomial

which is written in factored form. Note that the constants in the linear polynomials (like -3 and +1 in the above example) may be complex numbers in certain cases, even if all coefficients of the expanded form are real numbers. This is because the field of real numbers is not algebraically closed; however, the fundamental theorem of algebra states that the field of complex numbers is algebraically closed.
In school algebra, students learn to move easily from one form to the other (see: factoring).

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. Evaluation is sometimes performed more efficiently 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 will equal 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 a given 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, methods are given for solving a system of linear equations in several unknowns. To get a unique solution, the number of equations should equal the number of 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. This important subject is studied extensively in the area of mathematics known as linear algebra. 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. [2]