6.2: The Return of the Box

Earlier, we learned we can make a box from a piece of paper by cutting squares of side length [tex]$x$[/tex] from each corner and then folding up the sides. Let's say we now have a piece of paper that is 8.5 inches by 14 inches. The volume [tex]$V$[/tex], in cubic inches, of the box is a function of the side length [tex]$x$[/tex], where [tex]V(x) = (14 - 2x)(8.5 - 2x)(x)$[/tex].

1. Identify the degree and leading term of the polynomial. Explain or show your reasoning.

To identify the degree and leading term of the polynomial for the volume function [tex]$ V(x) = (14 - 2x)(8.5 - 2x)(x) $[/tex], we'll expand the expression step by step.

1. Expand the first two terms:
We start by expanding [tex]$(14 - 2x)(8.5 - 2x)$[/tex].

[tex]\[
(14 - 2x)(8.5 - 2x) = 14 \times 8.5 - 14 \times 2x - 2x \times 8.5 + 4x^2
\][/tex]

Calculate each term:
- [tex]$14 \times 8.5 = 119$[/tex]
- [tex]$14 \times 2x = 28x$[/tex]
- [tex]$2x \times 8.5 = 17x$[/tex]
- [tex]$4x^2$[/tex] is already calculated

So, the expansion becomes:

[tex]\[
119 - 28x - 17x + 4x^2 = 119 - 45x + 4x^2
\][/tex]

2. Expand with the third term:
Now multiply the result by [tex]$x$[/tex]:

[tex]\[
(119 - 45x + 4x^2)(x) = 119x - 45x^2 + 4x^3
\][/tex]

3. Identify the degree and leading term:
- The degree of the polynomial is the highest power of [tex]$x$[/tex], which is 3 because of the term [tex]$4x^3$[/tex].
- The leading term is the term with the highest power, which is [tex]$4x^3$[/tex].

Therefore, the degree of the polynomial is 3, and the leading term is [tex]$4x^3$[/tex].