Difference between revisions of "Polynomials in one variable"

Latest revision as of 10:42, 5 March 2024

Polynomial is an expression that consists of variables and coefficients. Polynomial involves the operations of addition, subtraction, multiplication, and exponentiation of variables. The word "polynomial" contains two words, namely, “poly" and “nomials”. "Poly" means many, and "nomials" means terms. Hence an expression containing many terms is called polynomials, having variables and coefficients.

Example: $3x+5$ , $x^{2}+4x-5$

In the expression $3x+5$ , $3$ is a coefficient and $x$ is a variable.

Degree of a Polynomial

The highest or greatest exponent of the variable in a polynomial is known as the degree of a polynomial. The degree is used to find the maximum number of solutions of a polynomial equation.

In the expression $x^{2}+4x-5$ , $x$ is a variable and its highest exponent is $2$ . Hence this is a second degree polynomial.

Standard form of Polynomial

The standard form of a polynomial refers to writing a polynomial in the descending power of the variable.

$5+2x+x^{2}$ can be written as $x^{2}+2x+5$ by placing the exponential terms in the descending order.

Types of Polynomials

Polynomials can be classified based on their degree and the number of terms

Based on the numbers of terms, there are mainly three types of polynomials which are provided below.

Monomial
Binomial
Trinomial

Monomial

Monomial is a type of polynomial with a single term.

Example: $x$ , $7y^{2}$

Bionomial

A Binomial is a type of polynomial that has two terms.

Example: $3x^{3}+5$ , $x+5$ x

Trinomial

A Trinomial is a type of polynomial that has three terms.

Example: $x^{2}+2x+5$

However, based on the degree of the polynomial, polynomials can be classified into 4 major types:

Zero Polynomial

Zero Polynomial is a polynomial whose value is zero.

Constant polynomial

Constant polynomial is a polynomial with just a constant and no variables.

Example: $5$

Linear polynomial

Linear polynomial is a polynomial with degree 1.

Example: $x+5$

Quadratic polynomial

Quadratic polynomial is a polynomial with degree 2.

Example: $x^{2}+2x+5$

Cubic polynomial

Cubic polynomial is a polynomial with degree 3.

Example: $3x^{3}+2x^{2}+2x+5$

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-Polynomials in one variable are algebraic expressions that are of the form <math>ax^n</math> where <math>n</math> is a non-negative (i.e. positive or zero) integer and <math>a</math> is a real number, called the coefficient of the term. The degree of a polynomial in one variable is the largest exponent in the polynomial.
+Polynomial is an expression that consists of variables and coefficients. Polynomial involves the operations of addition, subtraction, multiplication, and exponentiation of variables. The word "polynomial" contains two words, namely, “poly" and “nomials”. "Poly" means many, and "nomials" means terms. Hence an expression containing many terms is called polynomials, having variables and coefficients.
+Example: <math>3x+5</math> , <math>x^2+ 4x-5</math>
+In the expression <math>3x+5</math>,  <math>3</math> is a coefficient and  <math>x</math> is a variable.
+==Degree of a Polynomial==
+The highest or greatest exponent of the variable in a polynomial is known as the degree of a polynomial. The degree is used to find  the maximum number of solutions of a polynomial equation.
+In the expression <math>x^2+ 4x-5</math>,  <math>x</math> is a variable and its highest  exponent is <math>2</math>. Hence this is a second degree polynomial.
+==Standard form of Polynomial==
+The standard form of a polynomial refers to writing a polynomial in the descending power of the variable.
+<math>5+2x+x^2</math>can be written as <math>x^2+ 2x+5</math> by placing the exponential terms in the descending order.
+==Types of Polynomials==
+Polynomials can be classified  based on their degree and the number of terms
+Based on the numbers of terms, there are mainly three types of polynomials which are provided below.
+*Monomial
+*Binomial
+*Trinomial
+===Monomial===
+Monomial is a type of polynomial with a single term.
+Example: <math>x</math> , <math>7y^2</math>
+===Bionomial===
+A Binomial is a type of polynomial that has two terms.
+Example: <math>3x^3+5</math> , <math>x+5</math>x
+'''Trinomial'''
+A Trinomial is a type of polynomial that has three terms.
+Example: <math>x^2+ 2x+5</math>
+However, based on the degree of the polynomial, polynomials can be classified into 4 major types:
+===Zero Polynomial===
+Zero Polynomial is a polynomial whose value is  zero.
+===Constant polynomial===
+Constant polynomial is a polynomial with just a constant and no variables.
+Example:  <math>5</math>
+===Linear polynomial===
+Linear polynomial is a polynomial with degree 1.
+Example: <math>x+5</math>
+===Quadratic polynomial===
+Quadratic polynomial is a polynomial with degree 2.
+Example: <math>x^2+ 2x+5</math>
+===Cubic polynomial===
+Cubic polynomial is a polynomial with degree 3.
+Example: <math>3x^3 +2x^2+ 2x+5</math>