Problem 83 - College Algebra

Question

Is every rational function a polynomial function? Why or why not? Does a true statement result if the two adjectives rational and polynomial are reversed? Explain.

Step-by-Step Solution

Answer

No, every rational function is not a polynomial function because polynomial functions do not have variables in their denominator. However, every polynomial function is a rational function because every polynomial can be represented as a ratio where the denominator is 1.

Step 1: Defining Rational and Polynomial Functions

Rational functions are the ratio of two polynomial functions. The general form of a rational function is \(R(x) = \frac{p(x)}{q(x)}\), where \(p(x)\) and \(q(x)\) are polynomial functions and \(q(x)\neq0\). Polynomial functions are a sum of terms involving powers of the same variable or variables. The general form of a polynomial function is \(P(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_2x^2+ a_1x + a_0\), where \(a_n\neq0\).

Step 2: Rational Functions as Polynomial Functions

In the general form of a rational function, a rational function can only be a polynomial function if the denominator \(q(x)\) is 1. If \(q(x)\) is anything else, then the rational function is not a polynomial function because polynomial functions do not have variables in the denominator.

Step 3: Polynomial Functions as Rational Functions

In the general form of a rational function, if \(p(x)\) is a polynomial, and \(q(x)\) is 1, then every polynomial function is a rational function. This is because dividing any polynomial by 1 does not change the polynomial.