Degree of Polynomial

Polynomials are a fundamental concept of mathematics which is essential for the understanding of equations. The highest power of the polynomial expression is known as the Degree of Polynomial. 

In this article, we will learn what the Degree of Polynomial is, its importance and ways it can be used to solve mathematical problems. 

Introduction

The highest power of the variable in a polynomial equation is called the "Degree of the Polynomial''. It shows us how many terms there are in the equation must be a whole number, not a fraction or negative number.

The degree affects how the polynomial behaves and what it looks like. For example, a polynomial with a degree of 2 will always have a curved, parabolic shape, while one with a degree of 3 will form a cubic shape. Overall, it's a crucial concept in mathematics that provides us valuable information about the polynomial.

Finding the degree of a Polynomial

A polynomial is a mathematical expression that combines variables and coefficients by using only addition, subtraction, and multiplication. The variables and coefficients in a polynomial can be either real or complex numbers. A polynomial with only one term is called a monomial, while a polynomial with multiple terms is called a multinomial.

Polynomials are used in different areas of mathematics, such as algebra, calculus, and numerical analysis, and have practical applications in fields like modeling physical systems and solving optimization problems.

For example: A polynomial will look like this: p(x) = 3a5 + 4a2 + 3a5 + 2a4 + 5 + 6a3 + 3 + 8a

Steps to find the degree of polynomial are as follows: 

• Step 1: Combining the terms with equal powers. 

( 3a5 + 3a5 ) + 2a4 + 6a3 + 4a2 + 8a + (5+3)

• Step 2: We will then ignore all the coefficients: 

a5 + a5 + a4 + a3 + a2 + a1 + a0

• Step 3: Sorting the variable on the basis of their powers

a5 + a5 + a4 + a3 + a2 + a1 + a0

• Step 4: Finally, the largest power of the polynomial will be the degree of the polynomial. 

a5 + a5 + a4 + a3 + a2 + a1 + a0

Here, the term “a5" has the highest degree. Which means the degree of the polynomial is 5. 

Types of Polynomial

Based on degree, number of terms and coefficients, Polynomials are classified into several types. Here are some common classifications: 

1. Constant Polynomial: Polynomial with a degree 0. For ex: 8 or -8 etc. 
2. Linear Polynomial: Polynomial with a degree 1. For ex: 5x + 3 or -8x etc
3. Quadratic Polynomial: Polynomial with degree 2. For ex: 7x2 or 5x2 etc
4. Cubic Polynomial: Polynomial with degree 3. For ex: 8x3 or 6x3 etc. 
5. Quadratic Polynomial: Polynomial with degree 5. For ex: 4x4 or 7x4 etc. 

Applications of Degree of Polynomials

• Factorization: 

Degree of Polynomials is used to solve factorization of Polynomials. Polynomial of degree 2 can be broken down into two linear terms while a polynomial of degree 3 can be broken down to 3 linear terms. 

• Finding Zeros of Polynomials: 

It is also used to find the number of zeros, or roots of a polynomial equation. It is also useful in determining the x intercepts of a Polynomial, which are the points where the polynomials intercept the x axis.

• Graphical Polynomials: 

The degree of Polynomials is useful in finding the shape of a polynomial function such as straight line or the curve. This is useful for graphical polynomials and visualizing their behavior. 

• Geometry:

Degree of Polynomials is useful in finding solutions to problems in geometry. It can help us find the equation of parabolic path of a projectile.

Apart from this, it is very helpful in understanding polynomial interpolation and polynomial approximation. 

Importance

How to find whether the polynomial expression is homogenous or not? Degree of each term in the polynomials plays a very vital role as by evaluating each term we can know the polynomial expression is homogenous or not. 

For instance: Considering the polynomial expression 5x4 + 3xy4 + 7x4.

To know if the above expression is homogenous or not we will find the degree of each term. 

If the degrees of each term is equal then the expression is known as homogenous else it will be called non homogenous. 

As in the above example we can see that every term has a degree of 4 which means it is a homogeneous polynomial. 

Examples

• x4 + 5x6 + 7x2 + 3x3 : 

Degree of the polynomial is 6. 

• 8x2 + 7x2 + 7x3 

Degree of the polynomial is 3. 

• 4

Degree of the polynomial is 0.