What Are Polynomials?

Polynomials are algebraic expressions combinations of variables and coefficients. Arithmetic operations such as addition, subtraction, multiplication, and non-negative integer exponentiation of variables can be performed on polynomials but division by a variable cannot be performed on a polynomial. Polynomials are expressions containing constants, variables, and exponents that are combined using arithmetic operations.

Example- 3x^2+2y+6 is a polynomial expression where, 

Constant- 6

Variable- x and y

Exponent- 2 in 3x^2, 1 in 2y, and 0 in 6

Polynomials are a vital part of the “language” of algebra in mathematics. Polynomial is made up of two terms, Poly which means many, and Nominal means terms. Its classification is done on the basis of the number of terms present in the expression as monomial, binomial, and trinomial. These types of polynomials can be combined using some arithmetic operations such as addition, subtraction, multiplication, and division but are not divisible by a variable.

Degree of a Polynomial

The degree of a polynomial is the highest degree of a monomial within a polynomial or largest exponent of the variable. 

Example- 5x^4+8x^3+2x+6

Here, the degree of polynomial expression is 4. Each part of a polynomial in an equation is called a term. In the above expression, the number of terms is 4. Terms are 5x^4,8x^3,2x,6.

Types of Polynomials

How can you remember the names of the types of polynomials? Think about cycles! 

Monocycle(one wheel), Bicycle(two wheels) and Tricycle(three wheels)

Similarly, 

  • Monomial

A monomial is a type of polynomial expression that contains only one term, and the single term should be a non-zero term. example- 6, 4x, 8y^3, -2xy

  • Binomial

A Binomial expression contains exactly two terms. It can be formed by the sum or difference between two or more monomials. example- 3-2x, xy^4+5x

  • Trinomial

A trinomial expression contains exactly three terms. example- 5x^3+4y-8, 2y^3-3y+5

Standard Form of Polynomial Expressions

The standard form for writing a polynomial is to put the terms with the highest degree of the variable first. For example, to write a given polynomial 8+3x^3+5x+2x^2 standard form, we will first check the degree of a polynomial. Here, the highest degree is 3. Next, we will check for a term with a degree less than 3, i.e., 2, then a term with a degree less than 2,i.e 1 and 0. 8+3x^3+5x+2x^2 in standard form can be written as 3x^3+2x^2+5x+8.

Arithmetic Operations on Polynomials

Basic mathematical operations can be performed on polynomials just as they are done on numbers. If polynomials are added, subtracted, or multiplied, then the result is another polynomial and when polynomials are divided, then the result is a rational expression. Basic  operations on polynomial can be given as:

  • Addition of polynomials

The addition of polynomials is the same as the addition of numbers, the only difference is that we have to pair up like terms i.e; terms of the same variable and power, and then add them up.

  • Subtraction of polynomials

The subtraction of polynomials is also similar to the subtraction of two numbers. Only first we have to align like terms of the polynomial and then subtract them.

  • Multiplication of polynomials

The multiplication of polynomials follows the commutative property, distributive property, associative property, etc using the rules of exponents.

  • Division of polynomials

The division of polynomials is when we divide a given polynomial by another polynomial which is generally of a lesser degree in comparison to the degree of the dividend.

What is Special About Polynomials?

1. The Resulting Degree of the polynomial after addition and subtraction of polynomials is always the same.

2. The multiplication of polynomials results in a polynomial of the higher degree.

3. The division of polynomials may or may not result in a polynomial.

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