A constant polynomial is a type of polynomial that consists solely of a constant term without any variables. It is thus always a monomial.
Here are a few examples of constant polynomials:
f(x) = 7
g(x) = 8.5
h(x) = -12
p(x) = π
The standard form of a constant polynomial is expressed as
f(x) = c
Here, c is a real number.
The constant value c remains the same irrespective of the change in x. Thus, the domain of a constant polynomial consists of all real numbers (ℝ), and its range is a singleton set containing the constant value.
Note: If f(x) = 0, the constant polynomial is called a zero polynomial.
Degree
The constant polynomial f(x) can also be written as
f(x) = c ⋅ x0
Thus, the degree of a constant polynomial is 0, irrespective of the value of the constant.
Note: If c = 0, we obtain the zero polynomial, f(x) = 0, for which the degree is undefined.
Graphing
The graph of a constant polynomial represents a horizontal line parallel to the x-axis. Since the value of the polynomial remains constant, the graph stays at a fixed height above or below the x-axis based on the value of the constant.
For example,
The constant polynomial P(x) = 3 represents a straight line parallel to the x-axis at y = 3
Difference Between Constant and Zero Polynomials
While both constant and zero polynomials appear similar, there are some differences.
Basis
Constant Polynomial
Zero Polynomial
General Form
f(x) = c
f(x) = 0
Nature of Terms
The degree of each term is 0
The coefficient of each term is 0
Degree
0
Undefined.
Graph
Horizontal line parallel to the x-axis.
Coincides with the x-axis.
Domain
All real numbers.
All real numbers.
Range
{c}
{0}
Solved Examples
Determine if the following are constant polynomials. If they are, state their degree. a) f(x) = 4 b) g(x) = x + 5 c) h(x) = -9 d) k(x) = 0
Solution:
a) Given, f(x) = 4 It is a constant polynomial. Thus, the degree of f(x) is 0 b) Given, g(x) = x + 5 It is a non-constant polynomial. c) Given, h(x) = -9 It is a constant polynomial. Thus, the degree of h(x) is 0 d) Given, k(x) = 0 It is a non-constant polynomial.
Find the degree of P(x) = 7.5
Solution:
Given, P(x) = 7.5, a constant polynomial. Here, P(x) = 7.5 or P(x) = 7.5x0 Thus, the degree of P(x) is 0
If f(x) = -11, determine f(0), f(5), and f(-3)
Solution:
Given, f(x) = -11, a constant polynomial. Here, for any values of x, f(x) is always -11 for any input. Thus, f(0) = -11 f(5) = -11 f(-3) = -11
Find the degree of the polynomial h(x) = 5x0 – 0x1
Solution:
Given, h(x) = 5x0 – 0x1 ⇒ h(x) = 5, a constant polynomial. Thus, the degree of h(x) is 0
