In this post we will learn the concept of degree of polynomial and understand method to find the degree of any given algebraic equation.
The concept is important and need basic understanding of algebraic equations and its components like constant, variable and coefficient.
What is Degree of polynomial?
For any polynomial, the highest number of variable present in any entity is termed as the polynomial degree.
We know that polynomial is made of entity containing both constants and variable.
To find the degree of polynomial, we have to check all the given entities and find the one with the highest number of variable.
The total number of variable in that entity is termed as the degree of polynomial.
Degree of Polynomial Definition
Given below is the academic definition of degree of polynomial.
” Polynomial degree indicates the highest exponential power of variable present in the expression“
We already know that polynomial contains coefficients, variable and constants. But while finding degree, we just have to keep account of the power of variable and ignore the coefficients/constants.
How to find degree of polynomial?
In math, you will find infinite varieties of polynomial expression.
To find the degree of any possible polynomial, we need a robust method which works in all possible cases.
Given below are the steps which will help you figure out the degree of any possible algebraic expression.
Steps to find polynomial degree
(a) Identify the entities of given polynomial
Entities in algebraic expression is separated by addition/subtraction sign.
(b) For all the entities, separate the coefficients, constants and variables.
(c) Count the number of variables present in each entity.
(d) The highest variable of any entity is the degree of given polynomial.
I hope the steps are clear to you.
Given below are some examples of degree calculation of polynomials.
Examples of degree of Polynomial
Example 01
Find the degree of below polynomial.
\mathtt{9x\ +\ 15x^{2} +\ 6\ +\ 2x^{6}}
Solution
Follow the below steps:
(a) Identify the entities of polynomial.
Entities of polynomials are separated by addition/subtraction sign.
\mathtt{9x,\ 15x^{2} ,\ 6\ ,\ 2x^{6}} are the entities of above polynomial.
(b) Find exponent/power of variables in each entity
Exponent of 9x ⟹ 1
Exponent of \mathtt{15x^{2}} ⟹ 2
Exponent of 6 ⟹ 0
Exponent of \mathtt{2x^{6}} ⟹ 6
(c) Find Highest Power
Note that the highest power of given entities is 6.
Hence 6 is the degree of given polynomial
Example 02
Find the degree of polynomial
\mathtt{x^{2} \ +\ 5xy^{3} \ +\ 3y^{3}}
Solution
(a) Identify the entities
There are three entities in the polynomial.
\mathtt{x^{2} ,\ 5xy^{3} \ ,\ 3y^{3}} are the three entities.
(b) Find the power of variables of each entity
Exponent of \mathtt{x^{2}} ⟹ 2
Exponent of \mathtt{5xy^{3}} ⟹ 4
Exponent of \mathtt{3y^{3}} ⟹ 3
(c) Find the Highest power
The highest power of entity is 4.
Hence, degree of polynomial is 4.
Example 03
What is the degree of below polynomial?
3 + xy
Solution
(a) Find the entities of polynomial
3 and xy are the two entities.
(b) Find the power of each entity
Exponent of 3 ⟹ 0
Exponent of xy ⟹ 2
(c) Highest power of entity is 2
Hence the degree of polynomial is 2.
Example 04
Find the degree of polynomial
\mathtt{x^{3} +\ 4xy\ +\ 9x^{2} y\ +\ 3}
Solution
(a) Find all the entities
\mathtt{x^{3} ,\ 4xy\ ,\ 9x^{2} y\ ,\ 3} are the four entities of algebraic expression.
(b) Find the power of variables in each entity.
Exponent of \mathtt{x^{3}} ⟹ 3
Exponent of 4xy ⟹ 2
Exponent of \mathtt{9x^{2} y} ⟹ 3
Exponent of 3 ⟹ 0
(c) Highest power of polynomial is 3.
Hence degree of polynomial is 3.
Example 05
\mathtt{x^{2} +yx\ +\ zx\ +\ x^{3} yz\ }
Solution
(a) Find number of entities
are the entities present in the polynomial.</p>
<p style="font-size:18px">
<strong>(b) Find power of variables in each entity</strong>
Exponent of [latex] \mathtt{x^{2}} ⟹ 2
Exponent of yx ⟹ 2
Exponent of zx ⟹ 2
Exponent of \mathtt{x^{3} yz\ } ⟹ 5
(c) The highest power of exponent is 5.
Hence, the degree of polynomial is 5.
What does degree of polynomial signifies?
The degree of polynomial tells the number of possible solution for any given equation.
Consider the below equation.
x + 3 = 0
Here the degree of polynomial is 1.
It tells that there is only one value of x for which the above equation satisfies. (i.e. left side = right side)
Similarly for equation, \mathtt{x^{2} +\ 4x\ +\ 4\ =0} .
The degree for polynomial is 2.
It tells that there are two possible solution of x for which the above equation satisfies.
Frequently asked Question – Degree of polynomial
(01) What is the degree of polynomial 0?
If we multiply any variable with constant 0, we will get 0.
For example
0 . x ⟹ 0
0 . xy ⟹ 0
So there can be any number of variable possible with 0.
Hence the degree of 0 is indefinite.
(02) Can you find degree of below polynomial?
\mathtt{2\sqrt{x} \ +\ 3}
Solution
The algebraic expression can also be written as:
\mathtt{2\ .x^{\frac{1}{2}} +\ 3}
There are two entities in the expression.
Power of \mathtt{2.x^{\frac{1}{2}}} ⟹ 1/2
Power of 3 ⟹ 0
The highest power if 1/2.
Hence, the degree of given polynomial is 1/2.
(03) Can you show some example of degree 3 polynomials
Given below is 3 degree polynomial
\mathtt{( a) \ xyz\ +\ 1}\\\ \\ \mathtt{( b) \ x^{3} +\ 2y}\\\ \\ \mathtt{( c) \ xy^{2} +\ x^{2}}