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Constant Polynomial

A constant polynomial is a polynomial that has no variable term — it's simply a number. It takes the form f(x) = c, where c is any real number, and its graph is a horizontal line.

A constant polynomial is a polynomial of degree 0, defined as f(x)=cf(x) = c where cc is a nonzero real number. Because there is no variable with a positive exponent, the output of the function is the same for every input value of xx. The special case f(x)=0f(x) = 0 is called the zero polynomial, which is typically considered to have no defined degree rather than degree 0.

Key Formula

f(x)=c,c≠0f(x) = c, \quad c \neq 0
Where:
  • f(x)f(x) = the output of the polynomial function for any input x
  • cc = a nonzero real constant (the value of the polynomial)

Worked Example

Problem: Determine whether f(x) = 7 is a constant polynomial, and find its degree and its value when x = 5.
Step 1: Identify the form of the polynomial. The function f(x) = 7 has no variable term — it can be written as 7x⁰, which is just 7.
f(x)=7=7x0f(x) = 7 = 7x^0
Step 2: Find the degree. The highest power of x that appears is 0, so the degree is 0.
degree=0\text{degree} = 0
Step 3: Evaluate at x = 5. Since there's no x in the expression, the output is always 7 regardless of the input.
f(5)=7f(5) = 7
Step 4: Confirm: f(x) = 7 is a constant polynomial of degree 0, and f(5) = 7.
Answer: Yes, f(x) = 7 is a constant polynomial with degree 0. Its value at x = 5 is 7.

Visualization

Why It Matters

Constant polynomials are the simplest members of the polynomial family, and recognizing them helps you classify polynomials by degree. In graphing, they produce horizontal lines, which appear frequently in real-world contexts — for example, a fixed fee, a speed limit, or any quantity that stays the same regardless of the input variable.

Common Mistakes

Mistake: Saying f(x) = 0 is a constant polynomial of degree 0
Correction: The zero polynomial f(x) = 0 is a special case. Most textbooks leave its degree undefined (or assign it −∞) rather than calling it degree 0. A constant polynomial of degree 0 requires c ≠ 0.
Mistake: Thinking a constant polynomial isn't really a polynomial
Correction: A single number like 5 is a valid polynomial — you can write it as 5x⁰. Polynomials don't have to contain a visible variable.

Related Terms