Chapter 4.1, Problem 21AYU

In Problems $15-26$, determine which functions are polynomial functions. For those that are, state the degree. For those that are not, state why not. Write each polynomial in standard form. Then identify the leading term and the constant term.

$g\left(x\right)=x^{2/3}-x^{1/3}+2$

To determine

Whether the function $g\left(x\right)=x^{2/3}-x^{1/3}+2$ is a polynomial or not. If it is a polynomial, state the degree and for those that are not, state why not. Write a polynomial in the standard form, identify the leading term and the constant term.

Answer to Problem 21AYU

Solution:

The function $g\left(x\right)=x^{2/3}-x^{1/3}+2$ is not a polynomial because the exponents are not non- negative integers.

Explanation of Solution

Given Information:

The function $g\left(x\right)=x^{2/3}-x^{1/3}+2$

A polynomial function in one variable is a function of the form

$f\left(x\right)=a_n{x}^n+a_{n-1}{x}^{n-1}+\mathrm{...........}+a_{1}x+a_0$

Where $a_n,a_{n-1},\mathrm{.....},a_1,a_0$ are constants, called the coefficients of the polynomial, $n\ge 0$ is an integer and $x$ is a variable. If $a_n\ne 0$, it is called the leading coefficient and n is degree of polynomial, $a_0$ is called the constant term.

In the function $g\left(x\right)=x^{2/3}-x^{1/3}+2$ the exponents are $\frac{2}{3}$ and $\frac{1}{3}$. The exponents are not integers. For polynomial, all exponents must be non-negative integers.

Therefore, the function $g\left(x\right)=x^{2/3}-x^{1/3}+2$ is not a polynomial because the exponents are not non- negative integers.