Degree of a Polynomial with Examples and FAQs

What do you mean by Degree of a Polynomial?

The degree of any polynomial is the greatest power of the variable term of that polynomial.

For example, look at the polynomial below:

Here, x is the variable and the highest power of x is 10.

Hence, the degree of the polynomial is 10.

This was an easy example as the polynomial is in only one variable.

Let us take an example of a multivariable polynomial and define its degree.

x^{3} y^{4}+y^{6}+x^{2} y^{3}+y^{2}

Here, the first term has a degree equivalent to the sum of exponents of x and y, i.e.,

3 + 4 = 7.

The second term has a degree equivalent to the exponent of y i.e., 6.

The third term again has a degree equivalent to the sum of exponents of x and y, i.e.,

2 + 3 = 5.

The last term has a degree equivalent to the exponent of y, i.e., 2.

Hence the degree of the first term is the highest and therefore the degree of polynomial is 7.

Some polynomials with specific types of degrees

Zero polynomial

The Polynomial consisting of only one term, zero, is known as zero polynomial. The degree of zero polynomial is undefined.

For example: 0,0 . x \text { and } 0 . x^{2} are some examples of zero polynomial.

Let us discuss the various degrees of a polynomial in detail.

Constant Polynomial – Degree ‘0’ 

The polynomial consisting of one term, any constant value without a coefficient, is known as a constant polynomial.

For example: 5,23, \sqrt{29} are some examples of constant polynomials.

The degree of such a polynomial is zero as:

5 . x^{0}=5, \text { as } x^{0}=1 Similarly, it can be shown for the other constants.

23 \cdot y^{0}=23

\sqrt{29} \cdot a^{0}=\sqrt{29}

Linear polynomial – Degree ‘1’

The polynomial with degree 1 is linear polynomial, i.e., the highest exponential power of the variable of this polynomial is 1. The polynomial can have one variable or multiple variables.

For example: 

20 x+5 ( Linear polynomial in one variable x)

2 x-z    (Linear polynomial in two variables x \text{ and }z)

x+y+z  (Linear polynomial in three variables x,y \text{ and } z)

The highest power of any variable in the above examples is ‘1’ only.

Quadratic polynomial – Degree ‘2’

The polynomial with degree 2 is quadratic polynomial, i.e., the highest exponential power of the variable of this polynomial is 2. This type of polynomial can have one variable or multiple variables.

For example:

x^{2}-10 x+2 (Quadratic polynomial in one variable x)

xy-y+22 (Quadratic polynomial in two variables x\text{ and }y)

x y+5 y z-6 z (Quadratic polynomial in three variables x,y \text{ and } z)

The exponent values of two variables in one term of the polynomial are added to find the degree in the last two examples, hence the degree is 2.

Cubic polynomial – Degree ‘3’

The polynomial with degree 3 is a cubic polynomial, i.e., the highest exponential power of the variable of this polynomial is 3. This type of polynomial can again have one variable or multiple variables.

For example:

5 x^{3}-14 x+9 (Cubic polynomial in one variable x)

x y^{2}-x^{2} y+12 (Cubic polynomial in two variables x\text{ and } y)

x y z+12 x^{3}-6 y z (Cubic polynomial in three variables x,y \text{ and } z)

The exponent values of two or three variables in one term of the polynomial are added to find the degree in the last two examples, hence the degree is 3.

What is the significance of the degree of a polynomial?

The degree of the polynomial helps to determine the number of zeros or roots of a polynomial function. By looking at the degree we can quickly analyze how many roots of a function exists. The number roots of a polynomial are equal to the degree of the polynomial.

For example: f(x)=12 x^{3}-14 x+8, will have three roots for which value of f(x)=0.

Example

Identify the degree of the polynomials given below:

x^{4}-14 x y^{2}+2

x^{3}-4 x y^{4}+21

Solution:

i. The degree of the polynomial given is 4 as the first term is the one with the highest exponent value and its degree is 4.

ii. The degree of the second term is the sum of the exponents of x and y, i.e.,

1 + 4 = 5, hence the degree of the polynomial is 5.