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Polynomial - Elementary properties of polynomials

Polynomial - Elementary properties of polynomials: Encyclopedia II - Polynomial - Elementary properties of polynomials

All polynomials have an expanded form, in which the distributive law has been used to remove all parentheses. (Some polynomials also have a factored form, in which parentheses appear.) In expanded form, a term of a polynomial is a part of the polynomial that includes only the operation of multiplication (where whole number powers are viewed as repeated multiplication). Every polynomial in expanded form is a sum of terms ...

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Polynomial: Encyclopedia II - Polynomial - Elementary properties of polynomials

Polynomial - Elementary properties of polynomials

Polynomials are classified by their degree and number of variables. The degree of a term in a polynomial is the sum of all of the exponents on all of the variables in that term. (A variable with no exponent is said to have an understood exponent of 1.) The degree of the polynomial is the largest degree of any one term.

Example:

3x(x − y) + z

is equivalent to the expanded form

3x² − 3xy + z.

In this expanded form, the second term is −3xy and its degree is 2. The polynomial is a second degree polynomial in three variables. If x = 10, y = 5, z = 100 then the evaluation of the polynomial is 250.

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

a_nxⁿ + a_n−1xⁿ⁻¹ + ... + a₂x² + a₁x + a₀.

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

A form suited for efficient evaluation of the polynomial is the Horner scheme

((...(a_nx + a_n−1)x + ... + a₂)x + a₁)x + a₀.

Every polynomial equation is equivalent to an equation in the form of a polynomial equal to zero. When written in this form, the degree of the equation is the degree of the polynomial, and the number of "unknowns" is the number of variables in the polynomial. A system of polynomial equations is a set of equations in which each variable takes on the same value in every equation. Systems of equations are usually grouped with a single open brace on the left.

In elementary algebra, methods are given for solving all first degree and second degree polynomial equations in one unknown. In general, the number of solutions equals the degree, though it is necessary to consider multiplicity.

In elementary algebra, methods are given for solving systems of linear equations in several unknowns. To get a unique solution, the number of equations must equal the number of unknowns. More powerful methods for solving several polynomial equations in several unknowns are given in linear algebra.

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