Chapter 4.1, Problem 10E

To determine

To describe: the error in analyzing the function and make a correction.

Answer to Problem 10E

The given function is not a polynomial function.

Explanation of Solution

Given information:

A function is given as

  $f\left(x\right)=2x^4+4x-9\sqrt{x}+3x^2-8$

Analyzed as

  $f\left(x\right)$ is a polynomial function.

The degree is 4 and $f\left(x\right)$ is a quartic function.

Leading coefficient is $2$ .

Concept used:

A polynomial function is of the form

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

Terms of a polynomial function should be arranged in descending order according to its degree to express it in a standard form and degree of each term should be a positive integer or whole number. The coefficients should be real numbers.

Leading coefficient of a polynomial function is the coefficient of the leading term.

Degree of the polynomial is the degree of leading term or the height degree in the polynomial function.

For the polynomial $f\left(x\right)=a_n{x}^n+a_{n-1}{x}^{n-1}+\mathrm{...}+a_{1}x+a_0$ the leading coefficient is $a_n$ and the degree is $n$ .

Calculation:

Consider the given function.

  $f\left(x\right)=2x^4+4x-9\sqrt{x}+3x^2-8$

The above function can be written as

  $f\left(x\right)=2x^4+4x-9x^{\frac{1}{2}}+3x^2-8$

Now, degree of each term is not a whole number and all coefficients are real.

So, the function is not polynomial function.