Polynomials

Polynomials are algebraic expressions consisting of variables and coefficients that can perform arithmetic operations.

Standard Definition of Polynomial

The standard definition of a polynomial should be writing the polynomial in descending order of the degree assigned to the variable. Let us take examples of polynomials, 7x + 10 + x². The highest degree assigned to a variable here is 2, and the following is 1. We also have a constant 10 whose variable’s degree is zero. Therefore, according to the standard definition of polynomial, it will be written as x² + 7x + 10.

Types of Polynomials

Polynomials are defined by the highest degree and power assigned to them, and based on that, primarily, there are three types of polynomials – monomials, binomials, and trinomials.

Monomials are the type of polynomial with a single variable or two terms. Examples of polynomial here can be x+7, y³-4, or 2x²+9. Similarly, a trinomial has three terms like 3x³-4x+7 or 4x-y-1. However, if we are considering the degree of a polynomial, then they are primarily categorized into 4 major types:

Zero or Constant Polynomial

Linear Polynomial

Quadratic Polynomial

Cubic Polynomial

In simple terms, any polynomial with zero as the highest degree, like the constant terms 3 or 5, is called zero polynomial. Polynomials with the highest degree as 1 and 2 are simultaneously called linear and quadratic polynomials. In a similar vein, the examples of polynomials with 3 as the highest degree are 6x³-7x+3, 8x³+9x+1.

Factorization of Polynomials

Factorization refers to the process of decomposing a polynomial expression into a simple form so that the final product has reduceable factors. After factorization, the coefficients of the polynomial and the factors are in the same domain as that of the actual polynomial. For factorization, you can follow various techniques to deduce the final answer. Some of the common formulas used to factorize a polynomial are as follows:

Common Factors Method where you deduct the common factors first

Grouping method where you break down the polynomial into simple expressions and then divide

Factoring by splitting the middle term is also used, although it is commonly followed in quadratic polynomials only

Factorization using algebraic identities is one of the most common methods of factorization of a polynomial

Polynomial Operations

The basic mathematical operations can also be performed on a polynomial.

Addition: The addition of 2 polynomials is quite simple as the basic rules remain the same. You need to pair the terms with the same degrees and add their coefficient. The addition rules do not change here, but the numbers become polynomials

Subtraction: Similar to addition, the rules of subtraction also do not vary. You simply need to pair the like terms and conduct the operation

Multiplication: Polynomials also follow real numbers’ basic commutative, associative, and distributive properties. You can pair the terms side by side to solve the multiplication

Division: Although the division operation is a little tricky, a polynomial can be divided by another polynomial only if one is of a lesser degree than the other

Important Tips for Solving Polynomials

You can only use the plus or minus sign to solve a polynomial and separate the identities

The degree or the power on the variables needs to be a whole number only

The whole numbers in a polynomial are known as constant and the alphabets are the variables

When you are doing addition or subtraction of Polynomials, the operations can only be conducted with like terms

Conclusion

Polynomials are fun mathematical operations that are used widely in various fields every day. If you are interested in a career in mathematics, polynomials are your friends. Understanding the aspects of how it is used can actually give you lots of insights as to how to solve them!