Problem 9 - College Algebra

Question

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

No, the function \( f(x) = \frac{x^{2}+7}{x^{3}} \) is not a polynomial function since it has terms with negative exponents.

Step 1: Rewrite the function

The first step is to simplify the function \( f(x) = \frac{x^{2}+7}{x^{3}} \) to a form that allows easier comparison to the definition of polynomial functions. This can be done by dividing both terms in the numerator by \( x^{3} \), yielding the function \( f(x) = \frac{x^{2}}{x^{3}} + \frac{7}{x^{3}} = \frac{1}{x} + \frac{7}{x^{3}} \)

Step 2: Compare to a polynomial function

Now the function is in a form that allows comparison to a polynomial function. A polynomial function would be in the form \( a_{n}x^{n} + a_{n-1}x^{n-1} + \ldots + a_{2}x^{2} + a_{1}x + a_{0} \), and each \( n \) must be a non-negative integer (including zero). If you review the function \( f(x) = \frac{1}{x} + \frac{7}{x^{3}} \), you can see that there are terms with negative powers of \( x \) which contravenes the definition of a polynomial function. Therefore, \( f(x) \) is not a polynomial function.

Key Concepts

Degree of Polynomial

The degree of a polynomial is a very important concept in understanding polynomial functions. The degree is defined as the highest power of the variable in the polynomial expression when it is written in its standard form. This means that when you write out the polynomial fully, with all terms ordered from the highest to the lowest power, the largest exponent indicates the degree.
For example, in the polynomial expression \( 3x^4 + 5x^2 - 7 \), the degree is 4 because this is the highest exponent of the variable \( x \).

• The degree determines many properties of the polynomial function, such as its end behavior and the number of potential solutions or roots.
• It's crucial to ensure that exponents are non-negative integers, as this is a defining feature of polynomials.

When you're analyzing a function, checking for the highest power of \( x \) helps us confirm the degree and thus whether the function meets polynomial criteria.

Polynomial Expression

A polynomial expression is one of the most basic building blocks in algebra and higher mathematics. It consists of variables, often denoted as \( x \), raised to various powers and multiplied by coefficients.
Each of these constituent parts of the polynomial is called a 'term'.

• Coefficients are typically real numbers and may appear in front of variables; for example, in \( 4x^3 \), 4 is the coefficient.
• Polynomial expressions are sums of these terms, such as \( 2x^3 + 3x^2 - x + 5 \).
• The variables' exponents in polynomial expressions must be non-negative integers.

These expressions are highly structured and governed by specific rules, like operations with polynomials following the laws of arithmetic.
Understanding the arrangement and operation of such expressions gives insight into handling polynomial functions, equations, and inequalities.

Non-negative Integer Powers

Non-negative integer powers refer to the exponents in a polynomial expression that cannot be negative or fractional; they must be whole numbers equal to or greater than zero. This is a defining feature of a polynomial function, and here's why it matters.

• When powers are non-negative, it means you are not dividing by the variable. For instance, in expressions like \( x^{-2} \), you would be dividing by \( x^2 \).
• This rule ensures that the function stretches indefinitely in both directions along the x-axis without interruptions or undefined regions other than vertical asymptotes.
• Having non-negative powers in all terms allows a function to properly be classified as a polynomial.

This characteristic is fundamental because polynomials have a wide range of applications in solving real-world problems and modeling different natural phenomena, all based on their predictable and smooth curves.