Polynomial Mean In Math-Definition, Degree, And Notation

Polynomials are algebraic expressions that contain indeterminate (variables that don’t have a specific value) and constants. You can think of polynomials as their own special language within mathematics. They are used to express numbers in almost every field of mathematics and are considered very important in certain branches of math, such as calculus. For example, 2x + 5 and

 x² + 3x + 6 are polynomials.in this article learn about polynomials, definitions, types of polynomials, and properties of polynomials.  

What is a polynomial?

Polynomials get their name from two words: Poly, meaning “many,” and nominal, meaning “terms.” So by definition, a polynomial is an expression made up of variables (like x or y), constants (like 3 or 5), and exponents (like 2 or 4) that are combined using mathematical operations such as addition, subtraction, multiplication, and division.

The number of terms in the expression determines whether it is classified as a monomial, binomial, or trinomial. Examples of constants, variables, and exponents are as follows:

• Constants. Example: 1, 2, 3, etc.
• Variables. Example: g, h, x, y, etc.
• Exponents: Example: 5 in x⁵ etc.

Polynomial Definition

Polynomials are algebraic expressions in which the exponents of all variables must be whole numbers. The exponents of the variables in any polynomial can never be negative integers. A polynomial comprises of constants and variables, however, we cannot perform division operations by a variable in polynomials.

For example:

Let’s understand this with an example: 3x² + 5. In this polynomial, there are certain terms we need to be familiar with. Here, x is known as the variable. The number 3, which is multiplied by x², has a special name; we denote it by the term “coefficient.” The number 5 is known as the constant. The power of the variable x is 2.

Notation of polynomial

The polynomial function is represented by P(x), where x is the variable. For example, P(x) = x²-5x+11. If the variable is denoted by a, then the function will be P(a).

Terms of a polynomial

Polynomial terms are the parts of the equation that are typically separated by either “+” or “-” signs. In other words, each section of a polynomial in an equation is its own term. For example, in the polynomial equation 2x² + 5x +4, there would be a total of 3 terms. The classification of a polynomial is based on how many terms are within it

Polynomial	Terms
P(x) = x3-2x2+3x+4	x3, -2x2, 3x and 4
P(x) = 3x2+5x+4	3x2,+5x,+4

The standard form of a polynomial

A polynomial in standard form is expressed by writing the terms with the highest degree first, followed by the terms of the next highest degree, and so on. The standard form of a polynomial is given by

f(x) = a_nxⁿ + a_n-1 x^n-1 + a_n-2 x^n-2 + … + a₁x + a₀, where x is the variable and ai are coefficients.

In another word, the standard form of a polynomial is when the polynomial is written in descending power of the variable.

For example: 3 + 5x + x² in standard form can be written as x² + 5x + 3.

Degree of Polynomial

Degree refers to the highest exponent of a polynomial’s individual terms. For example, in the equation 2x^3 + 5x^2 + 3x + 1, the “x” has the highest exponent, so this would be classified as a 3rd-degree polynomial. In general, the degree of a polynomial equation with one variable is just the largest exponent of that variable.

Like term and unlike terms of polynomial

Terms in a polynomial are considered like terms when they share the same variable and power. Terms that have different variables or different powers are known as unlike terms. So, if a polynomial has two variables, all terms with the same power of any one variable would be considered like terms. Let us understand these two with the help of the examples given below.

For example:

2x and 3x are like terms. Whereas, 3y⁴ and 2x³ are unlike terms.

Frequently Asked Question-FAQs