The degree of a polynomial is the highest power of the variable in a polynomial expression. To recall, a polynomial is defined as an expression of more than two algebraic terms, especially the sum (or difference) of several terms that contain different powers of the same or different variable(s). It is a linear combination of monomials. For example: 6x 4 + 2x 3 + 3
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A polynomial’s degree is the highest or the greatest power of a variable in a polynomial equation. The degree indicates the highest exponential power in the polynomial (ignoring the coefficients).
For example: 6x 4 + 2x 3 + 3 is a polynomial. Here 6x 4 , 2x 3 , 3 are the terms where 6x 4 is a leading term and 3 is a constant term. The coefficients of the polynomial are 6 and 2.
The degree of the polynomial 6x 4 + 2x 3 + 3 is 4.
Let’s take another example: 3x 8 + 4x 3 + 9x + 1
The degree of the polynomial 3x 8 + 4x 3 + 9x + 1 is 8.
We know that the polynomial can be classified into polynomial with one variable and polynomial with multiple variables (multivariable polynomial). As discussed above, the degree of the polynomial with one variable is the higher power of the polynomial expression. But, if a polynomial has multiple variables, the degree of the polynomial can be found by adding the powers of different variables in any terms present in the polynomial expression.
Let’s consider a polynomial expression with two variables, say x and y
(i.e) x 3 + 6x 2 y 4 + 3y 2 +5
The degree of the polynomial is 6.
This is because in the second term of the algebraic expression , 6x 2 y 4 , the exponent values of x and y are 2 and 4, respectively. When the exponent values are added, we get 6. Hence, the degree of the multivariable polynomial expression is 6.
So, if “a” and “b” are the exponents or the powers of the variable, then the degree of the polynomial should be “a + b”, where “a” and “b” are the whole numbers.
A zero polynomial is the one where all the coefficients are equal to zero. So, the degree of the zero polynomial is either undefined, or it is set equal to -1.
A constant polynomial is that whose value remains the same. It contains no variables. The example for this is P(x) = c. Since there is no exponent so no power to it. Thus, the power of the constant polynomial is Zero. Any constant can be written with a variable with the exponential power of zero. Constant term = 6 Polynomial form P(x)= 6x 0
A Polynomial is merging of variables assigned with exponential powers and coefficients. The steps to find the degree of a polynomial are as follows:- For example if the expression is : 5x 5 + 7x 3 + 2x 5 + 3x 2 + 5 + 8x + 4
(5x 5 + 2x 5 ) + 7x 3 + 3x 2 + 8x + (5 +4)
x 5 + x 3 + x 2 + x 1 + x 0
x 5 + x 3 + x 2 + x 1 + x 0
deg( x 5 + x 3 + x 2 + x 1 + x 0 ) = 5
Every polynomial with a specific degree has been assigned a specific name as follows:-
| Degree | Polynomial Name |
|---|---|
| Degree 0 | Constant Polynomial |
| Degree 1 | Linear Polynomial |
| Degree 2 | Quadratic Polynomial |
| Degree 3 | Cubic Polynomial |
| Degree 4 | Quartic Polynomial |
To find whether the given polynomial expression is homogeneous or not, the degree of the terms in the polynomial plays an important role. The homogeneity of polynomial expression can be found by evaluating the degree of each term of the polynomial. For example, 3x 3 + 2xy 2 +4y 3 is a multivariable polynomial. To check whether the polynomial expression is homogeneous, determine the degree of each term. If all the degrees of the term are equal, then the polynomial expression is homogeneous. If the degrees are not equal, then the expression is non-homogenous. From the above given example, the degree of all the terms is 3. Hence, the given example is a homogeneous polynomial of degree 3.
Some of the examples of the polynomial with its degree are:
Example: Find the degree, constant and leading coefficient of the polynomial expression 4x 3 + 2x+3.
Given Polynomial: 4x 3 + 2x+3.
Here, the degree of the polynomial is 3, because the highest power of the variable of the polynomial is 3.
Leading Coefficient is 4. Because the leading term of the polynomial is 4x 3 .
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The degree of a polynomial is defined as the highest power of the variable of its individual terms (i.e. monomials) with non-zero coefficients.
A quadratic polynomial is a type of polynomial which has a degree of 2. So, a quadratic polynomial has a degree of 2.
A third-degree (or degree 3) polynomial is called a cubic polynomial.
To find the degree of the given polynomial, combine the like terms first and then arrange it in ascending order of its power.
So, 5x 5 +7x 3 +2x 5 +9x 2 +3+7x+4 = 7x 5 + 7x 3 + 9x 2 + 7x + 7
Thus, the degree of the polynomial will be 5.
If a and b are the exponents of the multiple variables in a term, then the degree of a term in the polynomial expression is given as a+b. For example, x 2 y 5 is a term in the polynomial, the degree of the term is 2+5, which is equal to 7. Hence, the degree of the multivariate term in the polynomial is 7.
