Chapter 6.5, Problem 74PPS

a.

To determine

To find:thevalue of the time for which the polynomial equation $f\left(t\right)=0$ .

Answer to Problem 74PPS

  $x^3-12x^2+4x-48=0$ .

Explanation of Solution

Given:

A population of parasites in an experiment can be molded by $f\left(t\right)=t^3+5t^2-4t-20$ .

Calculation:

As the polynomial equation is $f\left(t\right)=t^3+5t^2-4t-20$ .

Now, to find the factor of the equation where $f\left(t\right)=0$ . Therefore,

  $\begin{array}{l}f\left(t\right)=t^3+5t^2-4t-20\left\{\text{given polynomial}\right\}\\ t^3+5t^2-4t-20=0\{\text{given }f\left(t\right)=0\}\\ t^2\left(t+5\right)-4\left(t+5\right)=0\{\text{taking }\left(t+5\right)\text{as common}}\\ \left(t+5\right)\left(t^2-4\right)=0\{\text{use distributive property of product}}\end{array}$

Either $\left(t+5\right)=0$ or $\left(t^2-4\right)=0$

When $\left(t+5\right)=0$ , then $t=-5$

When $\left(t^2-4\right)=0$ , then $t^2=4\Rightarrow t=\pm \sqrt{4}=\pm 2$

Hence, the value of time is $t=-5,-2,2$ .

b.

To determine

To find: the value of x by using the polynomial factor.

Answer to Problem 74PPS

2 days.

Explanation of Solution

Given:

A population of parasites in an experiment can be molded by $f\left(t\right)=t^3+5t^2-4t-20$ .

Calculation:

From part (a), the time is $t=-5,-2,2$ .

This means there are 3 results which are $t=-5,-2,2$ .

Since the days is not negative, so the time when population reach zero, is 2 days

Since, $\left(x^2+4\right)$ cannot be zero, therefore, $\left(x-12\right)=0$ is zero.

  $\Rightarrow x=12$

Hence, the value of $x=12$ .

c.

To determine

To check: that for any value of t is unreasonable.

Answer to Problem 74PPS

  $t=-5,-2$ .

Explanation of Solution

Given:

A population of parasites in an experiment can be molded by $f\left(t\right)=t^3+5t^2-4t-20$ .

Calculation:

From part (a), the time is $t=-5,-2,2$ .

This means there are 3 results which are $t=-5,-2,2$ .

Since the days is not negative, Hence, the value of $t=-5,-2$ is unreasonable.