Question

Dimensions: x by 3x by x3
Area of base = x(3x) = 3x2
Box 2: Dimensions: x by 4x – 1 by x3
Area of base = x(4x – 1) = 4x2 – x

Complete the statements about the number of terms in the polynomial representing the volume of each box.

Answer 1

Box #1:
Dimensions are x, 3x,  and x³.
Base area, A = x*(3x) = 3x².
Volume, V = (3x²)*(x³) = 3x⁵.
The volume is a 5-th order polynomial. It has only one term, which is the leading term.

Box #2:
Dimensions are x, (4x-1), and x³.
Base area, A = x*(4x-1) = 4x² - x.
Volume, V = (4x² - x)*(x³) = 4x⁵ - x⁴.
The volume is a 5-th order polynomial. It has two terms. 
It could be written as
V(x) = 4x⁵ - x⁴ + 0x³ + 0x² + 0x¹ + 0x⁰.
Only terms involving x⁵ and x⁴ have non-zero coefficients.

Answer 2

The highest power of a variable in any polynomial is called the degree of the polynomial. Total number of term in any one variable polynomial is always one more than of its degree. Total number of terms in box 1 is 1 and in box 2 is 2.

Dimensions are given for both the boxes that are follow:

$\rm{Box\;1}=x \times 3x \times x^3\\\rm{Box\;2}=x \times (4x-1) \times x^3$

Compute the volume for box 1.

$\begin{aligned}\rm{Volume}&= x \times 3x \times x^3\\&=3x^5\end{aligned}$

Thus, the volume of the box 1 has degree 5 polynomial and have only one term in it.

Compute the volume for box 2.

$\begin{aligned}\rm{Volume}&= x \times (4x-1) \times x^3\\&=4x^5-x^4\end{aligned}$

Thus, the volume of the box 2 has degree 5 polynomial and have only two terms in it.

To know more about polynomials, please refer to the link:

https://brainly.com/question/15301188