Chapter 6.5, Problem 70PPS

a.

To determine

To write:the polynomial equation using the and formula for the volume.

Answer to Problem 70PPS

  $x^3-12x^2+4x-48=0$ .

Explanation of Solution

Given:

A rectangular prism with dimensions $x-2,x-4,\text{and }x-6$ .

Calculation:

As the volume is the product of its length, width and height. Therefore,

  $l=x-2,b=x-4\text{and }c=x-6$ . Now,

  $\begin{array}{l}\left(x-2\right)\left(x-4\right)\left(x-6\right)=40x\left\{\text{given}\right\}\\ \left(x-2\right)\left(x^2-10x+24\right)=40x\left\{\text{using algebric identity}\right\}\\ x^3-10x^2+24x-2x^2+20x-48=40x\{\text{simplify}}\\ x^3-12x^2+44x-2x^2+20x-48=40x\{\text{add like terms}}\\ x^3-12x^2+4x-48=0\{\text{equate one side to 0}}\end{array}$

Hence, the polynomial equal is $x^3-12x^2+4x-48=0$ .

b.

To determine

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

Answer to Problem 70PPS

  $x=12$ .

Explanation of Solution

Given:

The given polynomial is $x^3-12x^2+4x-48=0$ .

Calculation:

Now, to find the factor of the given polynomial. Therefore,

  $\begin{array}{l}{x}^3-12x^2+4x-48=0\left\{\text{given polynomial}\right\}\\ x^3-12x^2+4x-48=0\left\{expand\right\}\\ x^2\left(x-12\right)+4\left(x-12\right)=0\{\text{taking }\left(x-12\right)\text{as common}}\\ \left(x^2+4\right)\left(x-12\right)=0\{\text{use distributive property of product}}\end{array}$

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 x is unreasonable.

Answer to Problem 70PPS

  $x=\pm 2i$ is unreasonable.

Explanation of Solution

Given:

The given polynomial is $x^3-12x^2+4x-48=0$ .

Calculation:

From part (b), there is a factor of $\left(x^2+4\right)=0$ , which is imaginary number.

Now, $\left(x^2+4\right)=0x^2=-4x=\pm \sqrt{4\times -1}=\pm \sqrt{2^2{i}^2}=\pm 2i$ .

Hence, the value of $x=\pm 2i$ is unreasonable.

d.

To determine

To find: the dimensions of the prism.

Answer to Problem 70PPS

  $10\text{unit},8\text{unit},6\text{unit}$ .

Explanation of Solution

Given:

A rectangular prism with dimensions $x-2,x-4,\text{and }x-6$ .

From part (b) $x=8$ .

Calculation:

The dimensions of the prism are given below:

Therefore, $x-2=12-2=10,x-4=12-4=8,\text{and 12}-6=6$ .

Hence, the dimension of the prism is $10\text{unit},8\text{unit},6\text{unit}$ .