Polynomial

Polynomial

When we think about polynomials , one question  arises in our mind.

  What is a polynomial???

  Yes, to know about polynomials we shall start with its definition. So, let's start:

 Definition:

 In mathematics, an expression consisting of variables (x,y,z...etc) and  coefficients ( known/unknown) , and that involves addition, subtraction, multiplication and non negative integral powers/exponents of variables ; is called a polynomial.

    A polynomial in a single variable  x is of the form: 

  a₀xⁿ+ a₁xⁿ⁻¹ +...+aₙ .

   Types of polynomial:

Polynomial Degree:

Zero polynomial: a polynomial of degree zero is called a zero polynomial or a constant.
   As for example 0, 1,2,... are zero polynomials.

 Linear polynomial: A polynomial of degree 1 is called a linear polynomial.

 As for example  2x, x+3, x/5,... are linear polynomials.

 Similarly, polynomials with degree 2,3,4,5 are called quadratic, cubic, quartic, quantic polynomials respectively.

  Also polynomials are named differently according to the number of terms.

  A polynomial with  single, dual and triple terms  are called monomial, binomial, trinomial respectively.

 A polynomial with real coefficients is called a real polynomial and a polynomial with complex coefficients is called a complex polynomial.

A polynomial with integer coefficients is called an integer polynomial.

 A polynomial with one variable is called a univariate polynomial.

 As for example ( x+2) is an univariate polynomial.

A polynomial with two variables is called a bivariate polynomial.

 As for example  (x+3y+60) is a bivariate polynomial.

A polynomial with more than one variables is called a multivariate polynomial.

 As for example  (4x+y+7z-50) is a multivariate polynomial.

  Polynomial terms:

 Homogeneous polynomial:

A polynomial having more than one variable and each term of the polynomial having same degree n, is called a homogeneous polynomial of degree n.

 example:

   (x²+ 5xy+y²)  and (x³+3x²y+3xy²+y³) are homogeneous polynomials of degree 2 and 3 respectively.

Complete polynomial:

    A polynomial without any zero coefficient is said to be complete polynomial ; otherwise it is incomplete polynomial.

Vanishing polynomial:

    A polynomial all of whose coefficients are zero is called a vanishing polynomial.

Monic polynomial:

    A monic polynomial is an univariate polynomial in which the leading coefficient (the non zero coefficient of highest degree) is equal to one.

      So, a monic polynomial is of the form:     xⁿ+ aₙ_₁xⁿ⁻¹ +...+ a₁x + a₀ .

  Polynomial formula:

 (1)  addition,subtraction, multiplication of two or more polynomials are also polynomial.

(2)  division of two polynomials may not be a polynomial.

This is the basic idea about a polynomials.

(3) Derivatives and integration of a polynomial are also polynomials. 

If you find out any incorrect information or know anything more about this , please write it in the comment section!