Introduction to Polynomials

Mathematics is a subject that demands a clear understanding and strong basics. Without a strong foundation, it will be challenging to understand the concepts incorporated in the subject, especially when the students move to the higher classes. Polynomials are one of the essential concepts in Maths. Hence, having a solid grasp of the concepts of this topic is necessary. 

What are Polynomials?

A polynomial is a mathematical expression made up of coefficients and variables, often known as indeterminates. Poly means many, and nominal means terms in Greek. Therefore, a polynomial means an expression with many terms. Polynomials include variables and exponents that are added or subtracted. The parts of polynomials are known as terms. We can perform mathematical operations such as addition, multiplication, subtraction, and positive integer exponents on polynomial equations, but not division using variables. Polynomials are classified as monomial, binomial, and trinomial based on the number of terms present in the expression. 

An example of a polynomial is 3x+1, where 3x and 1 are terms. 

Degree of Polynomials

The highest power of a variable in a polynomial expression is the degree of the polynomial. The degree of the polynomial with one variable is the highest power of that specific polynomial expression. However, imagine a polynomial has more than one variable. In that case, the degree of the polynomial can be determined by summing the powers of the various variables in any terms in the polynomial expression.

Types of Polynomials

The polynomials are classified into three types based on the total number of terms in that equation. They are as follows:

• Monomial – An expression with only one term is referred to as a monomial. The sole term in an expression needs to be non-zero for it to be a monomial. Some of the examples for monomials are 5x, 3, 6×4, -3xy.
  
• Binomial – A polynomial expression with precisely two terms is referred to as a binomial. Binomial can also be defined as the difference or sum of two or more monomials. Some examples of binomials are – 5x+3, 6×4+ 17x, xy2+xy, etc.
  
• Trinomial – An expression that has exactly three terms is called a trinomial. Few examples for trinomials are -8×4+2x+7, 4×4+ 9x + 7. 

These three types of polynomials can be combined to perform operations related to addition, subtraction, and multiplication. However, not divisible by variable. 

Polynomial Operations

Students can perform various operations such as addition, subtraction, multiplication, and division, just like they do with whole numbers. Some important factors to consider while doing these operations are given below.

• Addition of Polynomials – While adding polynomials, always add similar terms, which means adding terms with the same variable and power. Adding the polynomials always gives a polynomial of the same degree.
  
• Subtraction of Polynomials – The sole difference between adding and subtracting polynomials is the type of operation. So, to get the answer, just like in addition, subtract the like terms. It is worth noting that subtracting polynomials yields a polynomial of the same degree.
  
• Multiplication of Polynomials – When two or more polynomials are multiplied, the outcome is invariably a higher degree polynomial (unless one of them is a constant polynomial).
  
• Division of Polynomials – Polynomials can be divided as well, just like all the other operations. The division of two polynomials may or may not result in a polynomial.

Solving Linear and Quadratic Polynomials 

Basic algebra and factorisation ideas can be used to solve any polynomial. The first step in solving the polynomial equation is to set the right-hand side to 0. The polynomial solutions can be explained in two different methods:

• Solving Linear Polynomials- Finding the solution to linear polynomials is simple and straightforward. First, remove the variable term and set the equation to zero. Then the students can solve it as though it were a simple algebra problem.
  
• Solving Quadratic Polynomials- This is little different than solving a linear equation. Here, students must first rewrite the statement in descending order of degree to solve a quadratic polynomial. Then, equate the equation and apply polynomial factorization to acquire the solution to the problem. 

We hope this article on polynomials helped you. If you have any doubts, feel to reach out to us.

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