Question

sandra works in the shipping and logistics department at an online retailer. she and her team are 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 this worked example.

• \\(x^2 - 9x - 20\\)
• \\(x^2 - 3x - 2\\)
• \\(x^2 + 3x + 2\\)
• \\(x^2 + 9x + 20\\)

Explanation:

Set up the division problem

$$ \text{Area of the base} = \frac{\text{Volume}}{\text{Height}} = \frac{x^3 - 6x^2 + 7x + 6}{x - 3} $$

Perform polynomial division

$$
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$$

1. Divide \(x^3\) by \(x\):

$$ x^2 $$

Multiply and subtract:

$$ (x^3 - 6x^2) - x^2(x - 3) = -3x^2 $$

1. Bring down \(7x\), divide \(-3x^2\) by \(x\):

$$ -3x $$

Multiply and subtract:

$$ (-3x^2 + 7x) - (-3x)(x - 3) = -2x $$

1. Bring down \(6\), divide \(-2x\) by \(x\):

$$ -2 $$

Multiply and subtract:

$$ (-2x + 6) - (-2)(x - 3) = 0 $$

State the resulting quotient

$$ x^2 - 3x - 2 $$

Answer:

• (A) \(x^2 - 9x - 20\)
• (B) \(x^2 - 3x - 2\) (Correct answer)
• (C) \(x^2 + 3x + 2\)
• (D) \(x^2 + 9x + 20\)