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Polynomial Function

A polynomial function is a function whose output is determined by a polynomial expression in the input variable. It has the form f(x)=anxn+an−1xn−1+⋯+a1x+a0f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0, where the exponents are whole numbers and the coefficients are real numbers.

A polynomial function of degree nn is a function f:R→Rf: \mathbb{R} \to \mathbb{R} defined by f(x)=anxn+an−1xn−1+⋯+a1x+a0f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0, where nn is a non-negative integer, each aia_i is a real number, and an≠0a_n \neq 0. The degree of the polynomial function is the highest power of xx that appears with a nonzero coefficient. Polynomial functions are continuous and smooth, meaning their graphs have no breaks, holes, or sharp corners.

Key Formula

f(x)=anxn+an−1xn−1+⋯+a1x+a0f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0
Where:
  • nn = the degree of the polynomial (a non-negative integer)
  • ana_n = the leading coefficient (must not be zero)
  • a0a_0 = the constant term
  • xx = the input variable

Worked Example

Problem: Given the polynomial function f(x)=2x3−5x2+3x−4f(x) = 2x^3 - 5x^2 + 3x - 4, find f(3)f(3).
Step 1: Substitute x=3x = 3 into the function.
f(3)=2(3)3−5(3)2+3(3)−4f(3) = 2(3)^3 - 5(3)^2 + 3(3) - 4
Step 2: Evaluate each power of 3.
=2(27)−5(9)+3(3)−4= 2(27) - 5(9) + 3(3) - 4
Step 3: Multiply the coefficients by the computed powers.
=54−45+9−4= 54 - 45 + 9 - 4
Step 4: Combine the terms from left to right.
=14= 14
Answer: f(3)=14f(3) = 14

Visualization

Why It Matters

Polynomial functions model a wide range of real-world situations — from the trajectory of a thrown ball (quadratic) to revenue and cost models in business (cubic or higher). They are among the most well-behaved functions in mathematics, which makes them central to calculus, where you'll use them to approximate more complicated functions through techniques like Taylor polynomials.

Common Mistakes

Mistake: Including negative or fractional exponents and still calling the result a polynomial function.
Correction: Every exponent in a polynomial function must be a non-negative integer. Expressions like x−2x^{-2} or x1/2x^{1/2} disqualify a function from being polynomial.
Mistake: Confusing the degree of the function with the number of terms.
Correction: The degree is the highest exponent, not the count of terms. For example, 4x5+14x^5 + 1 has degree 5 but only two terms.

Related Terms