Chapter 2.8, Problem 1MRPS

Solution

a.

To write: A polynomial equation that you can use to find the value of x .

The polynomial equation would be $x^3+6x^2+5x-180=0$ .

Given information:

The volume of the rectangular prism shown is 180 cubic inches.

  

Formula used:

Volume of rectangular box is length times width times the height of the box.

Calculation:

The volume of the box would be:

  $\begin{array}{c}V=x\left(x+1\right)\left(x+5\right)\\ =x\left(x\left(x+5\right)+1\left(x+5\right)\right)\\ =x\left(x^2+5x+x+5\right)\\ =x\left(x^2+6x+5\right)\\ =x^3+6x^2+5x\end{array}$

Since volume of the box is 180 cubic inches, so replace V with 180 as:

  $\begin{array}{c}180=x^3+6x^2+5x\\ 0=x^3+6x^2+5x-180\end{array}$

Therefore, the required polynomial equation would be $x^3+6x^2+5x-180=0$ .

b.

To identify: All the possible rational solutions of the equation in part (a).

The possible rational solutions of the equation would be $\pm 1,\pm 2,\pm 3,\pm 4,\pm 5,\pm 6,\pm 9,\pm 10,\pm 12,\pm 15,\pm 20,\pm 30,\pm 36,\pm 45,\pm 60,\pm 90,\pm 180$ .

Given information:

The volume of the rectangular prism shown is 180 cubic inches.

  

Formula used:

As per Rational Zero Theorem, the possible rational zeros for any function with integer coefficients can be obtained by dividing factors of leading coefficient of highest degree term with factors of the constant term as:

  $\frac{p}{q}=\frac{\text{Factor of constant term }{a}_0}{\text{Factor of leading coefficient term }{a}_n}$

Calculation:

In the equation $x^3+6x^2+5x-180=0$ , the leading coefficient is 1 and the constant term is $-180$ .

Factors of 1 are $\pm 1$ and factors of $-180$ are $\pm 1,\pm 2,\pm 3,\pm 4,\pm 5,\pm 6,\pm 9,\pm 10,\pm 12,\pm 15,\pm 20,\pm 30,\pm 36,\pm 45,\pm 60,\pm 90,\pm 180$ .

  $\begin{array}{c}\frac{p}{q}=\pm \frac{1}{1},\pm \frac{2}{1},\pm \frac{3}{1},\pm \frac{4}{1},\pm \frac{5}{1},\pm \frac{6}{1},\pm \frac{9}{1},\pm \frac{10}{1},\pm \frac{12}{1},\pm \frac{15}{1},\pm \frac{20}{1},\pm \frac{30}{1},\pm \frac{36}{1},\pm \frac{45}{1},\pm \frac{60}{1},\pm \frac{90}{1},\pm \frac{180}{1}\\ =\pm 1,\pm 2,\pm 3,\pm 4,\pm 5,\pm 6,\pm 9,\pm 10,\pm 12,\pm 15,\pm 20,\pm 30,\pm 36,\pm 45,\pm 60,\pm 90,\pm 180\end{array}$

Therefore, the possible rational solutions of the equation would be $\pm 1,\pm 2,\pm 3,\pm 4,\pm 5,\pm 6,\pm 9,\pm 10,\pm 12,\pm 15,\pm 20,\pm 30,\pm 36,\pm 45,\pm 60,\pm 90,\pm 180$ .

c.

To use: The synthetic division to find a rational solution of the equation. Show that no other real solutions exist.

The only real solution of the equation would be $x=4$ .

Given information:

The volume of the rectangular prism shown is 180 cubic inches.

  

Calculation:

By checking possible zeros in the given function, it can be seen that $x=4$ is a zero of the function. Now, use synthetic division as shown below:

  $\begin{array}{l}4\overline{)\begin{array}{l}165-180\\ \downarrow 440180\end{array}}\\ 110450\end{array}$

Because $x=4$ is a zero of the function, so the function can be written $f\left(x\right)=\left(x-4\right)\left(x^2+10x+45\right)$ .

Check for the solution of the quadratic equation using the determinant formula $b^2-4ac$ :

  $\begin{array}{c}D=b^2-4ac\\ ={10}^2-4\left(1\right)\left(45\right)\\ =100-180\\ =-80\end{array}$

Since the determinant is a negative number, therefore, there will be no real solutions for the quadratic equation and the only solution would be $x=4$ .

d.

To determine: What are the dimensions of the prism?

The dimensions of the prism are $4$ inches by $5$ inches by $9$ inches.

Given information:

The volume of the rectangular prism shown is 180 cubic inches.

  

Calculation:

Using $x=4$ , the other sides would be:

  $\begin{array}{c}x+1\Rightarrow 4+1\\ \Rightarrow 5\\ \\ x+5\Rightarrow 4+5\\ \Rightarrow 9\end{array}$

Therefore, the dimensions of the prism are $4$ inches by $5$ inches by $9$ inches.