Question

What is the domain of a polynomial?

Short answer

Answer: The domain of a polynomial function is all real numbers, represented as Domain = ℝ.

Step-by-step solution

Understand the concept of domain

The domain of a function is the set of all possible input values (independent variable) for which the function is defined. In other words, the domain represents the values of the independent variable for which the function produces a valid output.

Recognize a polynomial

A polynomial is a mathematical expression that consists of variables (usually represented by x or other letters), coefficients (constants), and exponents. Polynomials may have multiple terms, and they can be represented in the general form: $$P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_2x^2 + a_1x + a_0$$ Where \(a_i\) are the coefficients, and \(n\) represents the degree of the polynomial.

Analyze the features of polynomial functions

Polynomials have a few key features that distinguish them from other functions: 1. They are continuous, which means there are no breaks or gaps in the graph of the polynomial functions. 2. They are smooth, which means that the graph of the polynomial functions has no sudden turns or sharp corners. 3. The highest exponent is always a whole number, making the polynomial function always defined for real numbers.

Determine the domain of a polynomial function

Given the features of polynomial functions mentioned in Step 3, you can conclude that since they are continuous, smooth, and always defined for real numbers, their domain consists of all real numbers. So, the domain of a polynomial is: $$\text{Domain} = \mathbb{R}$$