Degree of Polynomial

Degree of a polynomial is defined as the highest power of the variable in the polynomial expression. A polynomial is defined as an algebraic expression that consists of variables and coefficients on which we can perform various arithmetic operations such as addition, subtraction, and multiplication, but we cannot perform division operations by a variable. (4x+3) and (x²+2x+5) are examples of polynomial expressions. So, the degree of these polynomial expressions is 1 and 2, respectively.

Degree of Polynomial is a very important topic for classes 9 and 10. So, in this article, you will learn about the degree of various types of polynomials like constant polynomial, zero polynomial, etc.

What is Degree of Polynomial?

While checking the degree of polynomial, we need to check the highest power of the variables and ignore their coefficients.

For example, (3x⁴+5x²+8) is a polynomial expression, and 3x⁴, 5x², and 8 are its terms. 3x⁴ is the leading term, and 8 is the constant term.

The coefficients of the given polynomial are 3 and 5. The degree of the given polynomial is 4 since it is the highest exponent of the variable in the given polynomial expression.

We know that the general form of a polynomial is axⁿ + bx^n-1 + cx^n-2 +... + 1. So, its degree is n as the highest power of x in the given polynomial expression is n. The image added below shows a polynomial of degree n along with its coefficient and the constant term.

Degree of Polynomial Definition

Degree of a polynomial is defined as the highest power of the variable in the polynomial expression.

Check: Polynomial Formula

Degree of Zero Polynomial

A polynomial is said to be a zero polynomial if the coefficients of all the variables are equal to zero. Let the zero polynomial be f (x) = 0. Now, we can write it as f(x) = 0x⁰, f(x) = 0x¹, f(x) = 0x², f(x) = 0x³, etc. So, we can say that the degree of zero polynomial is undefined. Sometimes it is defined as negative (-1 or -∞).

Check: Zeros of Polynomial

Degree of Constant Polynomial

A polynomial is said to be a constant polynomial if its value remains the same. Let the constant polynomial be P(x) = c, or we can write it as P(x) = Cx⁰ since the value of x0 is 1. So, we can say that the degree of constant polynomial is zero.

Example: P(x) = 13 = 13x⁰.

So, the degree of P(x) is zero.

Degree of Polynomial with more than one variable

If a polynomial has more than one variable, then its degree is calculated by adding the exponents of each variable.

Example: Calculate the degree of the polynomial 13x⁴ + 8x³y² + 7x²y+11xy.

Solution:

The given polynomial expression is 13x⁴ + 8x³y² + 7x²y+11xy.

Now, let's calculate the degree of each term.

13x⁴ has a degree of 4 since the power of x is 3.

8x³y² has a degree of 5 since the power of x is 3 and the power of y is 3. So, by adding the exponents of x and y, we get 5. 

7x²y has a degree of 3 since the power of x is 2 and the power of y is 1. So, by adding the exponents of x and y, we get 3. 

11xy has a degree of 2 since the power of both x and y is 1. So, by adding the exponents of x and y, we get 2. 

The largest degree out of these is 5, so the degree of the given polynomial expression is 5.

Classification of Polynomials Based on its Degree

Every polynomial has been assigned a name depending on the degree of the polynomial expression.

The following are some polynomial expressions depending on the degree of a polynomial, with examples.

Check: Types of Polynomials

How to find the Degree of Polynomial?

The following are the steps to determine the degree of a polynomial expression. Now, let us find the degree of the polynomial expression 7x⁴+6x³-2x⁴+12x²+9x+3

Step 1: Combine all the like terms of the given polynomial expression, where like terms are the terms that have the same variables and powers. Here, 7x4 and -2x4 are like terms.

(7x⁴-2x⁴)+6x³+12x²+9x+3 = 5x⁴+6x³+12x²+9x+3.

Step 2: Ignore the coefficients of all the variables.

x⁴+x³+x²+x¹+x⁰

Step 3: Now arrange all the variables in the descending order of their powers, i.e., from the greatest exponent to the least.

x⁴+x³+x²+x¹+x⁰

Step 4: Now identify the largest power of the variable x, as the degree of a polynomial expression is the highest exponent of the variable.

So, the degree of the given polynomial expression is 4.

Degree of Polynomial Applications

The following are some important applications of the degree of a polynomial:

• With the help of the degree of a polynomial, we can determine the maximum solutions of the given function.
• It also helps to determine the maximum number of times a function crosses the x-axis on a graph when graphed.
• It also helps to determine whether the given polynomial expression is homogeneous or not. By evaluating the degree of each term of the polynomial, we can determine the homogeneity of a polynomial expression.
• If the degrees of all terms of the given polynomial are equal, then it is said to be homogeneous; otherwise, it is non-homogeneous. For example, in 3x²+4xy+y², the degree of each term is 2. Therefore, the given polynomial expression is a homogeneous polynomial of degree 2.

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Degree of polynomial function examples

Example 1: Determine the degree, constant and leading coefficient of the polynomial expression 7x⁴−8x³+2x+5.

Solution:

Given Polynomial Expression = 7x⁴−8x³+2x+5

The highest exponent of variable x = 4

So, the degree of the given polynomial expression = 4

The leading coefficient of the polynomial is the coefficient with the highest exponent. 

So, the leading coefficient of given the polynomial expression = 7

Constant = 5

Example 2: Determine the degree of the polynomial expression 2x⁴+6x⁵−x³+3x²+x⁶+9.

Solution:

Given polynomial expression = 2x⁴+6x⁵−x³+3x²+x⁶+9

The polynomials are not arranged from greatest exponent to least. So, let us arrange them in descending order of their exponents first.

So, the obtained expression = x⁶+6x⁵+2x⁴−x³+3x²+9

Now, the highest exponent of the variable x = 6

So, the degree of the given polynomial expression = 6.

Example 3: Find the degree and constant of the polynomial expression 3x⁸−16x⁵+21x²−7x.

Solution:

Given polynomial expression = 3x⁸−16x⁵+21x²−7x

The highest exponent of the variable x = 8

So, the degree of the given polynomial expression = 8

The constant of the given polynomial expression = 0

Example 4: Determine the degree, constant and leading coefficient of the polynomial expression 13x³−15x²−11x+9.

Solution:

Given polynomial expression = 13x³−15x²−11x+9

The highest exponent of the variable x = 3

So, the degree of the given polynomial expression = 3

The leading coefficient of the polynomial is the coefficient with the highest exponent.

So, the leading coefficient of given the polynomial expression = 13

Constant = 9.

Example 5: Calculate the degree of polynomial 4x³ + 7x³y¹ + 11x²y³+17xy²+21y³.

Solution:

The given polynomial expression is 4x³ + 7x³y¹ + 11x²y³+17xy²+21y³.

Now, let's calculate the degree of each term.

4x³ has a degree of 3 since the power of x is 3.

7x³y¹ has a degree of 4 since the power of x is 3 and the power of y is 1. So, by adding the exponents of x and y, we get 4.

11x²y³ has a degree of 5 since the power of x is 2 and the power of y is 3. So, by adding the exponents of x and y, we get 5.

17xy² has a degree of 3 since the power of x is 1 and the power of y is 2. So, by adding the exponents of x and y, we get 3.

21y³ has a degree of 3 since the power of y is 3.

The largest degree out of these is 5, so the degree of the given polynomial expression is 5.