Degree of a Polynomial

Degree of a Polynomial
Go back to  'Polynomials'

Look at the polynomial function given below:

nth degree polynomial

For an nth degree polynomial function with real coefficients and the variable is represented as x, having the highest power n, where n takes whole number values.

Let's learn in detail about degree of polynomial and how to find degree of a polynomial.

In this mini lesson, you will learn about degree of a polynomial definition, degree of a polynomial example, degree of a polynomial function, degree of zero polynomial, degree of constant polynomial and cubic polynomial example.

Check out the interesting examples and try your hand at a few interactive questions at the end of the page.

Lesson Plan


What Is the Definition of Degree of a Polynomial?

Degree of a Polynomial Definition 

Degree of a polynomial is the greatest power of a variable in the polynomial equation. 

To determine the degree of a polynomial function, only terms with variables are considered to find out degree of any polynomial.

The highest exponential power of the variable term in the polynomial indicates the degree of that polynomial.


How to Find the Degree of a Polynomial? 

Consider the polynomial

\(p\left( x \right):2{x^5} - \frac{1}{2}{x^3} + 3x - \pi \)

The term with the highest power of x is \(2{x^5},\) and the corresponding (highest) exponent is 5. Therefore, we will say that the degree of this polynomial is 5. Thus, the degree of a polynomial is the highest power of the variable in the polynomial. We can represent the degree of a polynomial by \({\rm{Deg}}\left( {p\left( x \right)} \right)\). Some examples:

\[\begin{array}{l}{\rm{Deg}}\left( {{x^3} + 1} \right) = 3\\{\rm{Deg}}\left( {1 + x + {x^2} + {x^3} + ... + {x^{50}}} \right) = 50\\{\rm{Deg}}\left( {x + {\pi ^3}} \right) = 1\end{array}\]

Note from the last example above that the degree is the highest exponent of the variable term, so even though the exponent of π is 3, that is irrelevant to the degree of the polynomial.

Degree of a Zero Polynomial

When all the coefficients are equal to zero, the polynomial is considered to be a zero polynomial. So, the degree of the zero polynomial is either undefined, or defined in a way that is negative(-1 or -∞)

Degree of a Constant Polynomial

A constant polynomial (P(x) = c) has no variables. Since there is no exponent so no power to it. Thus, the degree of constant polynomial is zero.

Example: For 6 or 6x0, degree = 0 

Degree of a Polynomial With More Than One Variable

The degree of a polynomial with more than one variable can be calculated by adding the exponents of each variable in it.

Example: 5x3 + 6x2y2 + 2xy

  • 5x3 has a degree of 3 (x has an exponent of 3)
  • 6x2y2 has a degree of 4 (x has an exponent of 2, y has 2, and 2+2=4)
  • 2xy has a degree of 2 (x has an exponent of 1, y has 1, and 1+1=2)

The largest degree out of those is 4, so the polynomial has a degree of 4

 
tips and tricks
Tips and Tricks

In order to find the degree of any polynomial, you can follow these steps:

  1. Identify each term of the given polynomial.
  2. Combine all the like terms, the variable terms; ignore constant terms.
  3. Arrange those terms in descending order of their powers.
  4. And you'll get the term with highest exponent, which defines the degree of the polynomial.

What Are the Names of Polynomials Based on Degree? 

Let's classify the polynomials based on degree of a polynomial example.

Polynomials Degree Examples
 Constant Polynomial Polynomials with Degree 0   3
 Linear Polynomial Polynomials with Degree 1    x + 8
 Quadratic Polynomial Polynomials with Degree 2   3x2 - 4x + 7
 Quadratic Polynomial Polynomials with Degree 3   2x3 + 3x2 + 4x + 6
 Cubic Polynomial Polynomials with Degree 4   x4-16
 Quartic Polynomial Polynomials with Degree 5  4x5+ 2x3 - 20

What Is the Importance of Degree of a Polynomial?

  • To determine the most number of solutions that a function could have.
  • To determine the most number of times a function will cross the x-axis when graphed.
  • To check whether the polynomial expression is homogeneous, determine the degree of each term. When the degrees of the term are equal, then the polynomial expression is homogeneous& when the degrees are not equal, then the expression is said to be non-homogenous.

             4x3 + 3xy2+8y3

The degree of all the terms is 3

Hence, the given example is a homogeneous polynomial of degree 3

 
important notes to remember
Important Notes
  1. Degree of a polynomial with only one variable: The largest exponent of the variable in the polynomial.
  2. Degree of a polynomial with more than one variable: To find the degree of the polynomial, you first have to identify each term of that polynomial, so to find the degree of each term you add the exponents.
  3. Degree of a rational expression: Take the degree of the top (numerator) and subtract the degree of the bottom (denominator).

  4. Degree of any polynomial expression with root such as 3√x is 1/2.

  5. You can have a look at exponents and exponent rules for basic understanding of exponents and operations on exponents.

Solved Examples

Example 1

 

 

Help Liza determine the degree and the leading coefficient of the following polynomial expression 5x2 - 20x - 20

Solved Example 3: Liza solving math problem

Solution

For the given polynomial expression 5x2 - 20x - 20

  • The highest exponent is 2, and so degree of the expression = 2
  • The coefficient with the highest exponent will be the leading coefficient of the expression, so the leading coefficient = 5
degree= 2, leading coefficient= 5
Example 2

 

 

Brook's teacher asked him to determine the degree of the polynomial 5x4 + 3x2 - 7x5 + x7

Solved Example 2: Brook solving math problem

Let's help Brook solving it step by step!

Solution

In order to find the degree of the given polynomial,

  1. Check each term of the given polynomial. All are like terms with x as variable.
  2. Arrange these terms in descending order of their powers, which gives x7 - 7x5 + 5x4+ 3x2
  3.  Term with the greatest or highest exponent is x7, so the degree of the polynomial is 7
Degree of polynomial is 7
Example 3

 

 

Find a fourth degree polynomial satisfying the following conditions:

  • has roots- (x-2), (x+5)
  • that is divisible by 4x2

Solution

We are already familiar with the fact that a fourth degree polynomial is a polynomial with degree 4

Also, we know that we can find a polynomial expression by its roots.

First condition

(x-2) (x+5) = x(x+5) - 2(x+5) = x2+5x-2x-10 = x2+3x-10

Second condition

(x2+3x-10)(4x2) = x2.4x2 + 3x.4x2 - 10.4x2  = 4x4+12x3-40x2

Thus, the required polynomial =  4x+ 12x3- 40x2

4x4+12x3-40x2
 
Challenge your math skills
Challenging Questions
  1. Determine the possible number of terms that a polynomial with degree 10 can have.

Interactive Questions  

Here are a few activities for you to practice.

Select/Type your answer and click the "Check Answer" button to see the result.

 
 
 
 

Let's Summarize

We hope you enjoyed learning about degree of a polynomial definition, degree of a polynomial example, degree of a polynomial function, degree of zero polynomial, degree of constant polynomial, cubic polynomial example with the practice questions. Now, you will be able to easily find answers to degree of a polynomial and knowing degree of a polynomial solutions.

The mini-lesson targeted the fascinating concept of degre of a polynomial. The math journey around degree of a polynomial starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that is not only relatable and easy to grasp, but will also stay with them forever. Here lies the magic with Cuemath.


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At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

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Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we at Cuemath believe in.


FAQs

1. What is the degree of a polynomial?

The degree of a polynomial is the highest degree of the variable term(with non-zero coefficients) in the polynomial.

2. What is the degree of a quadratic polynomial?

The degree of a quadratic polynomial is 2.

3. What is degree 3 polynomial?

Cubic polynomials are degree 3 polynomials.

4. What is the degree of the zero polynomial?

The degree of the zero polynomial is undefined.

5. What is the degree of polynomial 5√x ?

The degree of 5√x is 1/2.

6. What is the degree of the polynomial 5 √ 3?

The degree of the polynomial 5 √ 3 is zero.

7. Why is the degree of a polynomial important?

The degree in a polynomial function determines:

  • the maximum number of solutions that a function could have.

  • the maximum number of times a function crosses the x-axis on graphing it.
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