In Exercises 1–10, determine which functions are polynomial functions. For those that are, identify the degree. f(x)=x^1/3 −4x²+7

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1. Recall that a polynomial function is a function of the form $f\left(x\right)=a_n^{}{x}^n+a_{\mathrm{n-1}}^{}{x}^{\mathrm{n-1}}+...+a_0^{}$ where each exponent $n$ is a non-negative integer (0, 1, 2, ...).
2. Examine each term of the function $f\left(x\right)=x^{\frac{}{}}-4x^2+7$: the first term is $x^{\frac{}{}}$, the second term is $4x^2$, and the third term is the constant $7$.
3. Check the exponents of each term: the first term has an exponent of $\frac{}{}$ (which is a fraction), the second term has an exponent of $2$ (which is a non-negative integer), and the constant term can be considered as $x^0$.
4. Since the first term has a fractional exponent, it violates the definition of a polynomial function, which requires all exponents to be whole numbers (non-negative integers).
5. Therefore, conclude that $f\left(x\right)=x^{\frac{}{}}-\; 4x^2\; +\; 7$ is not a polynomial function.

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 exponents multiplied by coefficients. For example, f(x) = 3x^2 - 5x + 7 is a polynomial, but functions with variables under roots or with fractional exponents are not considered polynomials.

Exponents and Their Restrictions in Polynomials

In polynomial functions, the exponents of the variable must be whole numbers (0, 1, 2, ...). Fractional or negative exponents indicate the function is not a polynomial. For instance, x^(1/3) is not allowed in polynomials because 1/3 is not an integer.

Degree of a Polynomial

The degree of a polynomial is the highest exponent of the variable with a non-zero coefficient. It determines the general shape and behavior of the polynomial graph. For example, in f(x) = -4x^2 + 7, the degree is 2 because the highest power of x is 2.