Introduction to Polynomials

A polynomial is an algebraic expression formed by adding and subtracting terms. These terms can be constants, coefficients, variables, or a combination of all. While solving polynomials, we have to use introductory algebra and factoring methods to derive the roots of polynomial equations. Lower-degree polynomials like linear polynomials or quadratic polynomials can be easily solved by following simple steps. Finding the roots of a higher-order polynomial is a multi-step process.

What is a Polynomial?

The word polynomial means many terms– It refers to a variety of expressions that include constants, variables, and exponents. It is an algebraic expression consisting of variables and exponents that consists of the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables. Polynomials can be both simple and complex expressions with higher degrees. They are often written in a standard form with terms in higher degrees, followed by the term with the smaller exponent. The coefficient associated with the term with the highest power is called the leading coefficient. A polynomial that doesn’t contain any variable is called a constant.

Degree of Polynomials

The degree of a polynomial is the highest power of an algebraic expression. Determining the degree or order of polynomials is important for adding, subtracting, multiplying, or dividing polynomials.

Types of Polynomials:

Classifying polynomials requires determining their degrees and number of terms. Polynomials can be classified into different types depending upon their degree and the number of terms involved.

Types of polynomials based on the number of terms: Polynomials with the specific number of terms are named by putting a prefix in their names:

Monomial— Mono means one, and monomial is a polynomial with exactly one term. For example, 4, 3x, 4x²y, etc.
Binomial— Bi means two, and binomial is a polynomial with exactly two terms. For example, 4x+5.
Trinomial— Tri means three, and trinomial is a polynomial with exactly three terms. For example, 2x²y+4x+9.
The prefix of the word “polynomial” is poly which means many. However, the word polynomial can be used for all numbers of terms, including monomials.

Types of polynomials based on the degree

The highest value of the exponent in an expression is known as the degree of Polynomial or order of the polynomial. While finding the degree of the polynomial, the terms should be arranged in either ascending or descending order. Based on the degree of the polynomial, polynomials can be classified into four major types:

Zero or Constant Polynomial: A constant polynomial is a polynomial without a variable which means it has only a constant part. For example, 2, -5, etc. Since there is no variable in such polynomials, we can say it is a polynomial with a zero degree.
Linear Polynomial: The polynomial expression whose degree is one is called a linear polynomial—for Example, 2x+7 or 6y+3.
Quadratic Polynomial: The polynomial expression with the highest degree as two is called a quadratic polynomial. For example 2x² + 5 or 16y² + 7.
Cubic Polynomial: The polynomial expression whose degree is three is called a cubic polynomial. For example, 7x²y + 5x+2,