Chapter 4.8, Problem 54E

To determine

To find: The graph of a function based on the given information.

Answer to Problem 54E

The required polynomial equation is $y=-\frac{1}{5}\left(x+1.5\right)\left(x-0.5\right)\left(x-4.5\right)$ .

Explanation of Solution

Given information:

The given intervals of zeros are:

  $\begin{array}{c}-2<x<-1\\ 0<x<1\\ 4<x<5\end{array}$

Consider the given range of the zeros:

  $\begin{array}{c}-2<x<-1\\ 0<x<1\\ 4<x<5\end{array}$

Consider the equation:

  $y=a\left(x-x_1\right)\left(x-x_2\right)\left(x-x_3\right)$

Substitute $-1.5$ for $x_1$ from the interval $-2<x<-1$ , $0.5$ for $x_2$ from the interval $0<x<1$ , $4.5$ for $x_1$ from the interval $4<x<5$

  $y=-\frac{1}{5}\left(x+1.5\right)\left(x-0.5\right)\left(x-4.5\right)$

Graph the above equation:

  

It is seen from the above graph that the polynomial can have a maximum degree.

Hence, the required polynomial equation is $y=-\frac{1}{5}\left(x+1.5\right)\left(x-0.5\right)\left(x-4.5\right)$ .