What is the form of a polynomial function?

• A. ax + b
• B. a_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0
• C. c/x
• D. e^x + sin(x)

Answer: (B)

A polynomial function is defined as a mathematical expression that involves variables raised to whole number powers and multiplied by coefficients. The general form of a polynomial function is represented as a sum of terms, where each term consists of a coefficient multiplied by the variable raised to a non-negative integer exponent.

The correct answer clearly illustrates this concept by showing the summation of terms from the highest degree term down to the constant term. Each term is structured as ( a_n x^n ), where ( a_n ) represents the coefficients, ( x ) is the variable, and ( n ) is a non-negative integer indicating the degree of that term. The notation reflects the standard way to present polynomials, encompassing all the possible terms that can be included in a polynomial function.

In contrast, other options are not valid polynomial expressions. For example, one option involves a fraction with ( x ) in the denominator, indicating a rational function rather than a polynomial. Another involves exponential and trigonometric functions, which also do not conform to the structure of a polynomial. Thus, the detailed breakdown showcases how the correct answer is specifically aligned with the definition and form of polynomial functions.