Question

question
sandra works in the shipping and logistics department at an online retailer. she and her team
working on a new box design, where the height of the box is ( x - 3 ).
if the volume of the box is ( x^3 - 6x^2 + 7x + 6 ), what is the area of the base?
use long division to find the area expression, and then select the correct answer. for help, see
worked example

( \bigcirc x^3 - 9x - 20 )

( \bigcirc x^2 - 3x - 2 )

( \bigcirc x^2 + 3x + 2 )

( \bigcirc x^3 + 9x + 20 )

Explanation:

Step1: Recall Volume Formula

The volume \( V \) of a box is given by \( V=\text{Base Area}\times\text{Height} \). So, to find the base area, we divide the volume by the height. The height is \( x - 3 \) and the volume is \( x^{3}-6x^{2}+7x + 6 \). We use polynomial long division to divide \( x^{3}-6x^{2}+7x + 6 \) by \( x - 3 \).

Step2: Divide the Leading Terms

Divide the leading term of the dividend (\( x^{3} \)) by the leading term of the divisor (\( x \)): \( \frac{x^{3}}{x}=x^{2} \). Multiply the divisor \( x - 3 \) by \( x^{2} \): \( x^{2}(x - 3)=x^{3}-3x^{2} \). Subtract this from the dividend:

$$
LATEXBLOCK0
$$

Step3: Divide the New Leading Term

Now, divide the leading term of the new dividend (\( -3x^{2} \)) by the leading term of the divisor (\( x \)): \( \frac{-3x^{2}}{x}=-3x \). Multiply the divisor \( x - 3 \) by \( -3x \): \( -3x(x - 3)=-3x^{2}+9x \). Subtract this from the new dividend:

$$
LATEXBLOCK1
$$

Step4: Divide the New Leading Term

Divide the leading term of the new dividend (\( -2x \)) by the leading term of the divisor (\( x \)): \( \frac{-2x}{x}=-2 \). Multiply the divisor \( x - 3 \) by \( -2 \): \( -2(x - 3)=-2x + 6 \). Subtract this from the new dividend:

$$
LATEXBLOCK2
$$

So, the quotient is \( x^{2}-3x - 2 \), which is the area of the base.

Answer:

\( x^{2}-3x - 2 \) (corresponding to the option " \( x^{2}-3x - 2 \)")