Problem 115 - Algebra and Trigonometry

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, not every rational function is a polynomial function because rational functions can have negative powers or division by a variable which polynomials cannot. However, every polynomial function can be considered a rational function.

Step 1: Understanding Polynomial functions

A polynomial function is an expression consisting of variables and coefficients, which only employs the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. Examples can be \(x^2\), \(3x^2 + 2x + 1\) etc.

Step 2: Understanding Rational functions

A rational function is the ratio of two polynomials, with both polynomials in the ratio being defined by the rules of polynomial functions. Such functions look like this: \(f(x) = P(x)/Q(x)\), where both P and Q are polynomial functions.

Step 3: Comparing the two

Every polynomial could be seen as a rational function. For example the polynomial function \(P(x) = x^2\) can also be defined as a rational function \(f(x) = x^2/1\). However, not every rational function is a polynomial. For example the rational function \(f(x) = 1/x\) is not a polynomial because, based on the definition, polynomials do not have negative powers or division by a variable.

Step 4: Reversing the statement

If we reverse the adjectives, we say: 'Every polynomial function is a rational function'. This statement is true. Polynomials, such as \(x^2 + 2x + 1\), can be expressed as a ratio of two polynomials, such as \((x^2 + 2x + 1) / 1\).