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

Verified step by step guidance

1. Recall that a polynomial function is a function of the form \(h(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0\), where each exponent \(n, n-1, \ldots, 1, 0\) is a whole number (non-negative integer), and the coefficients \(a_i\) are real numbers.
2. Examine the given function: \(h(x) = 7x^3 + 2x^2 + \frac{1}{x}\). Notice that the first two terms, \$7x^3\( and \)2x^2$, have exponents 3 and 2, which are whole numbers.
3. Look closely at the term \(\frac{1}{x}\). This can be rewritten as \(x^{-1}\), which has an exponent of \(-1\), a negative integer.
4. Since polynomial functions cannot have negative exponents, the presence of \(x^{-1}\) means \(h(x)\) is not a polynomial function.
5. Therefore, \(h(x)\) is not a polynomial function, and identifying the degree does not apply in this case.

Key Concepts

Polynomial Functions

A polynomial function is a function that can be expressed as a sum of terms consisting of variables raised to non-negative integer powers multiplied by coefficients. It has the general form f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0, where n is a whole number and coefficients are real numbers.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial with a non-zero coefficient. It indicates the polynomial's order and affects the shape and behavior of its graph. For example, in 7x^3 + 2x^2 + 1, the degree is 3.

Non-Polynomial Terms

Terms with variables in the denominator or with negative or fractional exponents are not part of polynomial functions. For instance, the term 1/x can be rewritten as x^(-1), which disqualifies the function from being a polynomial because polynomials require non-negative integer exponents.