Problem 56 - College Algebra

Question

What is a polynomial function?

Step-by-Step Solution

Answer

A polynomial function is a mathematical function that can be expressed in the form f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_2 x^2 + a_1 x + a_0, where n is a nonnegative integer and a_0, a_1, a_2, ..., a_n are the coefficients. The degree of the polynomial is the highest power in the variable x.

Step 1: Define Polynomial Function

A polynomial function is a function that can be written in the form of a polynomial. It is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients.

Step 2: Explain the components of a polynomial function

The general form of a polynomial function is f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_2 x^2 + a_1 x + a_0. Here, x is the variable of the function, n is a nonnegative integer, and a_0, a_1, a_2, ..., a_n are the coefficients of the terms of the polynomial. The highest power of the variable x is called the degree of the polynomial.

Step 3: Illustrate with examples

For example, f(x) = 2x^3 - 3x^2 + x - 1 is a polynomial function of degree 3. Another example is g(x) = 5x - 7, which is a polynomial function of degree 1 (also called a linear function).