Problem 9 - College Algebra

Question

In Exercises \(1-10\), determine which functions are polynomial functions. For those that are, identify the degree. $$f(x)=\frac{x^{2}+7}{x^{3}}$$

Step-by-Step Solution

Answer

The function \(f(x) = \frac{x^{2}+7}{x^{3}}\) is not a polynomial function.

Step 1: Determine the nature of the function

Check if the function \(f(x) = \frac{x^{2}+7}{x^{3}}\) can be represented as a sum of monomials. Rewrite the function, \(f(x) = x^{-1}+7x^{-3}\). Upon rewriting, it is clear that the function has two terms: \(x^{-1}\) and \(7x^{-3}\), both of which have negative exponents.

Step 2: Verify the function's nature

Analyzing the rewritten function, it's clear that all the exponents are not nonnegative integers. Thus, the function \(f(x) = \frac{x^{2}+7}{x^{3}}\) is not a polynomial function.

Key Concepts

Understanding the Degree of a Polynomial

The degree of a polynomial is incredibly important as it tells us the highest power of the variable in the polynomial expression. Imagine you're stacking blocks, and you want to know how tall the highest stack is; that's much like identifying the degree of a polynomial. It is defined as the greatest exponent of any term in the polynomial. For a single variable polynomial like \( f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 \), where \( a_n \) is not zero, the degree is simply \( n \).

Finding the degree helps us predict the behavior of the polynomial's graph, such as the number of possible turning points and the behavior as the inputs grow large. In the given exercise, you were asked to determine the degree after simplifying the expression. However, since the presence of negative exponents disqualified it as a polynomial function, identifying the degree becomes irrelevant in this specific context.

Monomials and Their Role in Polynomials

Monomials form the building blocks of polynomial functions. A monomial is a single term consisting of a coefficient and a variable raised to a nonnegative integer power, like \( 7x^2 \). It's like a single LEGO piece in a larger structure. When you combine multiple monomials, adding or subtracting them, you get a polynomial.

By understanding monomials, you grasp the fundamental nature of polynomial functions. In the provided exercise, you're examining if the expression can be seen as a sum of monomials. The catch here is that monomials have nonnegative integer exponents—a key distinguishing feature—one that the terms after rewriting the initial expression do not satisfy.

The Implications of Negative Exponents

Negative exponents in an expression imply division by that variable raised to a positive exponent. Simply put,\( x^{-n} \) is the same as \( 1/x^n \), where \( n \) is a positive integer. It's a way of expressing how many times you divide by the variable. In regular polynomials, we stick to nonnegative integer exponents because they represent repeated multiplication. The occurrence of negative exponents, such as in the terms \( x^{-1} \) and \( 7x^{-3} \) from the exercise, is a sign that the function deviates from the standard definition of polynomial functions.

These negative exponents can lead to division by zero, creating undefined conditions for certain values of \( x \). That's a big no-no for polynomial functions, which have to be defined for all real numbers. This is a prime example of how negative exponents alert us to the presence of non-polynomial behavior in mathematical expressions.