We all have studied about polynomials in our school. Can you recall it? Any algebraic expression consisting of variables and coefficients are considered as polynomials. The word polynomial has been derived from Greek words, poly which signifies ‘many’ and nominal which means ‘terms’. In a nutshell, it means ‘many terms’. The expression composed of variables, functions, and constants in order to do various mathematical problems such as addition, subtraction and multiplication are also known as polynomials. Polynomials may have many terms but not infinite terms.
We all have studied polynomials of various degrees and their variables. Let us now discuss various types of polynomials and their various degrees.
Degree of Polynomial
The highest degree of a monomial within a binomial is referred to as the degree of a polynomial. Suppose that if p(x) is a polynomial in x the highest power of x in p(x) is the degree of a polynomial. For instance, take an example of an equation where 4x + 2 is a polynomial, thus the variable of x has the degree of 1. The following points discussed below mentions the types of degree of Polynomial.
1. Linear polynomial:
A polynomial containing one term is known as linear polynomial. Any polynomial which is expressed in the form of an equation where a and b are real numbers and a is not equal to 0, then the degree is known as linear polynomial. Let us take a few examples.
- 2x – 3
- 3x + 5
2. Quadratic polynomial:
A polynomial containing two terms is known as quadratic polynomial.The highest exponent of the variable in a quadratic polynomial is 2. The examples of a quadratic polynomial are mentioned below.
- y.y – 2
- 2x.x – 3
3. Cubic polynomial:
A polynomial containing three terms is known as a cubic polynomial. In a cubic polynomial, the highest exponent of the variable is 3. The examples of a cubic polynomial are mentioned below:
- 2x – x.x.x
- 5x.x.x + 6x + x
- 7x.x.x + 8x.x + 10x + 1
Types of Polynomials:
There are basically three types of polynomials which are namely- monomial, binomial, and trinomial. These types of polynomials are classified on the basis of the number of terms present in an expression. The parts of the equation which are generally separated by ‘+’ or ‘-‘ signs are regarded as the terms of the polynomials. Various types of polynomials are discussed below:
1. Monomial:
A polynomial containing only one term is said to be a ‘monomial’. For any term to be a monomial the single term should be a non-zero term. Few examples of monomials are mentioned below:
- 5x
- -3xy
- 3
2. Binomial:
A polynomial containing exactly two terms is said to be a ‘binomial’. The sum or difference between two or more monomials are considered as binomial. Few examples of binomials are mentioned below:
- -5x + 3
- -3x + 4
- +4 – 3x
3. Trinomial:
A polynomial containing exactly three terms is said to be a ‘trinomial’. A trinomial contains three terms and a variable in the expression. Few examples of trinomials are mentioned below:
- x.x + 2x + 20
- 2x.x + 4x + 5
- x.x + 4x + 25
Some Properties of Polynomials:
We shall now discuss some important properties of the polynomial along with its theorems. These theorems and properties are as follows:
1. Division Algorithm
P(x)= G(x) × Q(x) + R(X)
The equation mentioned above states that if polynomial P(x) is divided by a polynomial G(x) it results in the quotient Q(x) with remainder R(x).
2. Bezout theorem
P(a)= 0
If a polynomial P(x) is divisible by binomial x-a, then P(a) becomes equal to zero
3. Remainder theorem
P(a)= r
If P(x) is divided by (x-a) with remainder r, then the above equation is justified.
- The polynomial P which is divisible by polynomial Q signifies that every zero of Q is a zero of R.
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