Chapter 2.2, Problem 37E

Solution

To write: A polynomial of degree n and discuss its end behaviours.

The required polynomial function is $f\left(x\right)=3-x^5$ .

Given data: The polynomial is f is of degree 5 such that the end behaviour of the graph of function f is given by $f\left(x\right)\to +\infty$ as $x\to -\infty$ and $f\left(x\right)\to -\infty$ as $x\to \infty$ . Verify answer with graph of the function.

Method/Formula used:

An expression of the form $f\left(x\right)=\pm a_0\pm a_{1}x\pm a_2{x}^2\pm \cdots \pm a_n{x}^n$ is called the polynomial of degree n if and only if,

(i) $a_0,a_1,a_2,\dots ,a_n$ , are real numbers

(ii) n is a non-negative integer.

Calculation:

A polynomial of degree 5 can be written as: $f\left(x\right)=\pm a_0\pm a_n{x}^n$

Choose $a_0=3$ , $a_n=1$ , and $n=5$ .

Therefore, the required polynomial is $f\left(x\right)=\pm 3\pm 1x^5$ , so there are four different polynomials represented by the function $f\left(x\right)$ .

Consider polynomial $f\left(x\right)=3-x^5$ , then

(i) If $x\to -\infty$ ,

  $\begin{array}{c}f\left(-\infty \right)=3-{\left(-\infty \right)}^5\because {\left(-\infty \right)}^{\text{odd}\text{number}}=-\infty \\ =3-\left(-\infty \right)\\ =3+\infty \because \infty +a\left(\text{numbe}r\right)=-\infty \\ =+\infty \end{array}$

(i) If $x\to \infty$ ,

  $\begin{array}{c}f\left(\infty \right)=3-{\left(\infty \right)}^5\because {\left(\infty \right)}^{\text{odd}\text{number}}=\infty \\ =3-\infty \\ =-\infty \end{array}$

Therefore, the polynomial is f is of degree 5 such that $f\left(x\right)\to +\infty$ as $x\to -\infty$ and $f\left(x\right)\to -\infty$ as $x\to \infty$ is $f\left(x\right)=3-x^5$ .

The graph of the function $f\left(x\right)=3-x^5$ is shown in Fig. (1):

  

Interpretation: As clear from the graph the end behaviour of the graph of function f is $f\left(x\right)\to +\infty$ as $x\to -\infty$ and $f\left(x\right)\to -\infty$ as $x\to \infty$ . That is required.