Chapter 3, Problem 1RE

To determine

Whether the function $f\left(x\right)=4x^5-3x^2+5x-2$ is polynomial function, a rational function or neither and to state the degree if the function is polynomial function.

Answer to Problem 1RE

The function $f\left(x\right)=4x^5-3x^2+5x-2$ is a polynomial function and the degree of the polynomial function is 5.

Explanation of Solution

Definition used: A polynomial function in one variable is a function of the form $f\left(x\right)=a_n{x}^n+a_{n-1}{x}^{n-1}+\mathrm{...}+a_{1}x+a_0,a_n\ne 0$ where $a_n,a_{n-1},\mathrm{...},a_1,a_0$ are real numbers and n is a nonnegative integer. The domain of a polynomial function is the set of all real numbers.

The coefficients $a_n,a_{n-1},\mathrm{...},a_1,a_0$ are real numbers.

The leading term is $a_n{x}^n$; $a_n$ is called the leading coefficient.

The term $a_0$ is called constant term.

The degree of a polynomial function is the largest power of x that appears.

Calculation:

The given function is $f\left(x\right)=4x^5-3x^2+5x-2$.

The function $f\left(x\right)$ can be written as $f\left(x\right)=4x^5-3x^2+5x-2$ where 4 is the leading coefficient and $-2$ is the constant term.

The above mentioned function $f\left(x\right)=4x^5-3x^2+5x-2$ follows the definition of polynomial function.

Therefore, the function $f\left(x\right)=4x^5-3x^2+5x-2$ is a polynomial function.

From the polynomial function $f\left(x\right)=4x^5-3x^2+5x-2$, it is clear that the largest power of x that appear in $f\left(x\right)$ is 5.

Therefore, the degree of the polynomial function $f\left(x\right)=4x^5-3x^2+5x-2$ is 5.