Chapter 4.5, Problem 50E

(a)

To determine

To calculate: A polynomial expression to determine the value of $x$

Explanation of Solution

Given information:

Volume of the cube $=8\text{cubic}\text{centimetres}$

Length of the cube $=$ breadth of the cube $=$ height of the cube $=x-3\text{cm}$

Formula used:

Volume = length × breadth × height

Calculation:

  $\begin{array}{l}V=lbh\\ $ 8={(x-3)}^3$2=x-3\\ x=5\end{array}$

(b)

To determine

To calculate: The possible rational solutions for a).

Explanation of Solution

Given information:

Volume of the cube $=8\text{cubic}\text{centimetres}$

Length of the cube $=$ breadth of the cube $=$ height of the cube $=x-3\text{cm}$

Formula used:

Volume=length × breadth × height

Calculation:

  $\begin{array}{l}$ 8={(x-3)}^3$x^3-9x^2+27x+27=8\\ x^3-9x^2+27x-35=0\end{array}$

(c)

To determine

To calculate: The synthetic division to find a rational solution of the equation.

Explanation of Solution

Given information:

Volume of the cube $=8\text{cubic}\text{centimetres}$

Length of the cube $=$ breadth of the cube $=$ height of the cube $=x-3\text{cm}$

Calculation:

  $\begin{array}{l}$ 8={(x-3)}^3$x^3-9x^2+27x+27=8\\ x^3-9x^2+27x-35=0\end{array}$

$5$	$1$	$-9$	$27$	$-35$
		$5$	$-20$	$35$
	$1$	$-4$	$7$	$0$

(d)

To determine

To find: The dimensions of the cube

Explanation of Solution

Given information:

Volume of the cube $=8\text{cubic}\text{centimetres}$

Length of the cube $=$ breadth of the cube $=$ height of the cube $=x-3\text{cm}$

Since, $x=5$

So, the dimensions are $2,2,2$ .