Exponents and Polynomials

The algebraic expressions where the exponents are whole numbers are known as Polynomials. The term “polynomial” is rooted in the Greek word “poly” and “nominal” meaning “many” and “terms”, respectively. Therefore, Polynomials can have several terms; however, the terms cannot be infinite. It includes algebraic expressions comprising of the coefficients and variables, which are often known as the indeterminates. It is made up of variables, Exponents, and constants. Each part of a polynomial is known as “terms”. It is generally the difference or sum of exponents and variables. 

Variables are the terms or alphabets that help in representing an unknown and desired number or quantity or values. These alphabetic terms are used especially for algebraic expressions. For instance, in the equation y + 4 = 8, the term “y” is a variable; on the other hand, 4 and 8 are the constants. These constants can be used for securing the value of the given variable, using mathematical operations. The variables are of two types- Independent Variable and Dependent Variable. An example of an independent variable is z = a2. However, the equation y + 4 = 9 is an dependent variable. In the given independent variable z = a2, the exponent used is “2”. 

Now that we know what is Polynomial, let us discuss the degree of the polynomials. 

Degree

The degree of a polynomial can be understood by analyzing the highest exponent power of a term in a polynomial equation. Therefore, the degree of the polynomial can be explained through the following table:

Here-

• The constants are often known as Zero Polynomial, in the concepts of Polynomial Equations. 
• The polynomial equation with just the degree one is known as the Linear Polynomial. 
• The polynomial equation with degree two is known as Quadratic Polynomial.
• The polynomial equation with degree three is known as Cubic Polynomial.
• The Polynomial Equation with degree four is known as Quartic Polynomial.

Terms and Types

The polynomial equations when separated using mathematical operators are known as Terms of a polynomial. Therefore, each part of the polynomial equation is a term. 

There are three types of Polynomials. These polynomials are-

• Monomial Polynomial: The monomial polynomials comprise only one polynomial term. For example- 5x, x/3, 2y4, etc.
• Binomial Polynomial: The binomial polynomial comprises two polynomial terms. For example- 2y + 2, 2y4 + y3, etc. 
• Trinomial Polynomial: the Trinomial polynomials comprises of three polynomial terms. For example- y3+ y2 + 2y, y2 + 2y + 2, etc.

Zeros

The zeros of a polynomial are the real values of polynomial just so the value of the polynomial becomes zero. Let us understand this concept with the help of an example. 

Let there be two polynomial equations- 

P(a) = y + 2, and

P(b) = y – 2

For the above-given polynomial equations, the zeros of a polynomial are -2 and 2, respectively. 

This is calculated as:

=> y + 2 = 0 => y = 0 – 2 => -2

=> y – 2 = 0 => y = 0 + 2 => 2

It is not necessary for a zero of polynomial to be zero. However, the linear polynomial has one and only one zero of the polynomial.

Now, let us look at the several polynomial operations which are used for solving the polynomial equations.

Operations

While solving polynomial equations, one uses four types of Polynomial Operations. These operations are:

• Polynomial Addition- The addition of Polynomial is done by adding the like terms of a polynomial equation. These terms must have the same variable and the same exponent power. 
• Polynomial Subtraction- Subtraction of Polynomial is done by finding out the difference between the like terms in a polynomial equation. These terms must have the same variable and the same exponent power. 
• Polynomial Multiplication- The multiplication of polynomial is done with the help of the distributive law of multiplication. There is no provision of multiplying the like terms of a polynomial equation. 
• Polynomial Division- For a polynomial equation that has two or more terms, the long division method is used. However, the division of Polynomial may not always result in another polynomial equation. 

Conclusion

The algebraic expressions where the exponents are whole numbers are known as Polynomials. The term “polynomial” is rooted in the Greek word “poly” and “nominal” meaning “many” and “terms”, respectively. The degree of a polynomial can be understood by analyzing the highest exponent power of a term in a polynomial equation. The polynomial equations when separated using mathematical operators are known as Terms of a polynomial. There are three types of a polynomial- Monomial, Binomial, and Trinomial Polynomial. While solving all these types of polynomial, four polynomial operations are used. These operations are Addition of Polynomial, Subtraction of Polynomial, Multiplication of Polynomial, and Division of Polynomial. The zeros of a polynomial are the real values of polynomial just so the value of the polynomial becomes zero.