The degree of a polynomial is the highest power of the variable in the expression. A polynomial expression in standard form is written as P(x) = anxn + an – 1xn – 1 + … + a1x1 + a0
The degree of the polynomial 2x3 + 3x2 – 4x + 5 is 3 (the highest exponent of the given variable x is 3)
The degree of the polynomial 6x5 – x + 7 is 5 (the highest exponent of x is 5)
It is used to classify and analyze polynomials.
How To Find
The degree of a polynomial is based on the number of variables in the expression.
With One Variable
For polynomials with a single variable, the degree is the highest power of the variable with a non-zero coefficient.
For example,
Deg(x6) = 6
Deg (x2 + 2x + 1) = 2
For More Than One Variable
For multivariable polynomials, the degree is determined through the following steps:
Let us consider a polynomial with 2 variables, a and b:
P(a, b) = 4a2 + ab2 + a2b + 3b2
Arranging the Terms in Standard Form
Terms are arranged in descending order of their power
P(a, b) = 4a2 + a2b + ab2 + 3b2
Calculating the Degree of each Term
The degree of each term is calculated by summing the exponents of all variables in that term.
Here,
The term 4a2 has a degree of 2
The term a2b has a degree of 2 + 1 = 3
The term ab2 has a degree of 1 + 2 = 3
The term 3b2 has a degree of 2
Determining the Degree of the Polynomial
The term with the highest degree is the degree of the polynomial.
Since the highest degree among all terms is 3, the degree of P(a, b) is 3
Deg(P(a, b)) = 3
The degree of another polynomial 6a3b3 + 2ab2c2 + c4 is shown:
Constant Polynomial
A constant polynomial has a constant term with no variable. Thus, the degree of a constant polynomial is always 0. For example, f(x) = 5 has a degree of 0
Zero Polynomial
A zero polynomial contains no terms with non-zero coefficients. Thus, the degree of a zero polynomial is undefined.
Classification of Polynomials – Based on Degree
Here are the different types of polynomials based on their degree:
Polynomial
Degree
Example
Constant
0
P(x) = 9
Linear (1st-degree polynomial)
1
P(x) = 2x + 9
Quadratic (2nd-degree polynomial)
2
P(x) = x2 – 2x + 9
Cubic (3rd-degree polynomial)
3
P(x) = 5x3 + x2 – 2x + 9
Quartic (4th-degree polynomial)
4
P(x) = 3x4 + 5x3 – x + 4
Quintic (5th-degree polynomial)
5
P(x) = x5 + 5x – 8
Solved Examples
Determine the degree of the following polynomials: a) f(x) = 7x4 – 3x + 2 b) g(x) = x2 + 3x + 5 c) h(a, b) = 3a3b2 – 2a2b4 + ab d) k(x, y) = 5x2y3 – 4xy + 7 e) m(x, y) = 2x3 + 3xy2 + y4
Solution:
a) Given, f(x) = 7x4 – 3x + 2 Here, The term 7x4 has a degree of 4 The term -3x has a degree of 1 The constant term 2 has a degree of 0 Thus, the degree of f(x) is 4 b) Given, g(x) = x2 + 3x + 5 Here, The term x2 has a degree of 2 The term 3x has a degree of 1 The constant term 5 has a degree of 0 Thus, the degree of g(x) is 2 c) Given, h(a, b) = 3a3b2 – 2a2b4 + ab Here, The term 3a3b2 has a degree of 3 + 2 = 5 The term -2a2b4 has a degree of 2 + 4 = 6 The term ab has a degree of 1 + 1 = 2 Thus, the degree of h(a, b) is 6 d) Given, k(x, y) = 5x2y3 – 4xy + 7 Here, The term 5x2y3 has a degree of 2 + 3 = 5 The term -4xy has a degree of 1 + 1 = 2 The constant term 7 has a degree of 0 Thus, the degree of k(x, y) is 5 e) Given, m(x, y) = 2x3 + 3xy2 + y4 Here, The term 2x3 has a degree of 3 The term 3xy2 has a degree of 1 + 2 = 3 The term y4 has a degree of 4 Thus, the degree of m(x, y) is 4
Classify the following polynomials by their degree: a) p(x) = 2x + 5 b) q(x) = x3 – 4x2 + 2x – 1
Solution:
a) Given, p(x) = 2x + 5 Here, the degree of p(x) is 1 (as the highest exponent is 1) Thus, p(x) is a linear polynomial. b) Given, q(x) = x3 – 4x2 + 2x – 1 Here, the degree of q(x) is 3 (as the highest exponent is 3) Thus, q(x) is a cubic polynomial.
