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Chapter 2: Problem 5

Determine which functions are polynomial functions. For those that are, identify the degree. $$h(x)=7 x^{3}+2 x^{2}+\frac{1}{x}$$

Short Answer

Expert verified
The function \(h(x) = 7x^3 + 2x^2 + \frac{1}{x}\) is not a polynomial function due to the term \(\frac{1}{x}\). Therefore, it doesn't have a degree.

Step by step solution

01

Identify Individual Terms

The function \(h(x)\) is made up of three terms: \(7x^3\), \(2x^2\), and \(\frac{1}{x}\). Each of these needs to be examined to determine if it contributes to the formation of a polynomial function.
02

Decide If Each Term is part of a Polynomial Function

To decide if each term is part of a polynomial function, consider the exponent of the variable. In both \(7x^3\) and \(2x^2\), the variable \(x\) is raised to a positive integer which is allowed in polynomial functions. However, in the term \(\frac{1}{x}\), the variable \(x\) is in the denominator of the term, this is equivalent to \(x^{-1}\), which is not an integer greater than or equal to 0. Therefore, \(\frac{1}{x}\) is not a term that would be found in a polynomial function.
03

Determine if Function is a Polynomial Function

Since one of the terms of the function \(h(x)\), specifically \(\frac{1}{x}\), does not match the definition of a term in a polynomial function, the entire function \(h(x)\) is not a polynomial function.

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