Chapter 4.1, Problem 8E

To determine

Whether the given function is polynomial function or not. Also write the polynomial function is standard form and write the type, degree, and leading coefficient of the polynomial function.

Answer to Problem 8E

The given function is not a polynomial function.

Explanation of Solution

Given information:

A function is given as

  $h\left(x\right)=3x^4+2x-\frac{5}{x}+9x^3-7$

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.

  $h\left(x\right)=3x^4+2x-\frac{5}{x}+9x^3-7$

The above function can be written as

  $h\left(x\right)=3x^4+2x-5x^{-1}+9x^3-7$

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

So, the function is not polynomial function.