Chapter 2.2, Problem 4E

Solution

To check: A given function is a polynomial, write it in standard form, determine its degree and leading coefficient.

The function $f\left(x\right)=6x+8x^4-3$ is a quartic polynomial function with degree 4 and leading coefficient $8$ .

Given data: The given function is $f\left(x\right)=6x+8x^4-3$ .

Definition used: An expression $f\left(x\right)$ of the form $f\left(x\right)=a_n{x}^n+a_{n-1}{x}^{n-1}+a_{n-2}{x}^{n-2}+\cdots +a_3{x}^3+a_2{x}^2+a_{1}x+a_0$ is called a polynomial of degree n in standard form if,

( i) $a_0,a_1,a_2,a_3,\cdots a_n$ are real numbers ( some of them may be zero),

(ii) n is a non-negative integer.

(iii) $a_0$ is a constant.

The coefficient of highest power of variable x is known as the leading coefficient. Here, $a_n$ is the leading coefficient.

Calculation:

The given function $f\left(x\right)=6x+8x^4-3$ can be written as,

  $f\left(x\right)=8x^4+0\cdot x^3+0\cdot x^2+6\cdot x-3$ ... (1)

By the definition of a polynomial function the function $f\left(x\right)=6x+8x^4-3$ is a polynomial function and expression (1), is written in the standard form. The highest power of x is 4 so $f\left(x\right)$ is a quartic polynomial function. The coefficient of $x^4$ is 8, therefore, $8x^4$ is the leading term and $8$ is the leading coefficient.