A square metal sheet has a side length of 3x^3 inches. A square cutout with a side length … — ACT

A square metal sheet has a side length of $3x^3$ inches. A square cutout with a side length of $(x^3 - 2)$ inches is removed from the center of the sheet. Which of the following polynomial expressions represents the area, in square inches, of the remaining metal sheet?

A
$8x^6 - 4x^3 + 4$
B
$8x^5 + 4x^3 - 4$
C
$8x^6 + 4$
$8x^6 + 4x^3 - 4$ Answer
E
$2x^6 + 4x^3 - 4$

Answer

The expression

8x^6 + 4x^3 - 4

represents the remaining area of the metal sheet.

To find the remaining area, subtract the area of the square cutout from the total area of the metal sheet. The total area is the square of the side length

3x^3

, which is

(3x^3)^2 = 9x^6

. The area of the cutout is the square of

(x^3 - 2)

, which is

(x^3 - 2)^2 = x^6 - 4x^3 + 4

. Subtracting the cutout area from the total area yields

9x^6 - (x^6 - 4x^3 + 4) = 9x^6 - x^6 + 4x^3 - 4 = 8x^6 + 4x^3 - 4

. This matches the correct expression.

Step-by-Step Solution

Calculate the total area of the square metal sheet.

Area_{total} = (3x^3)^2 = 9x^6

The area of a square is the square of its side length. According to the power of a product rule,

(ab)^n = a^n b^n

, and the power of a power rule,

(x^a)^b = x^{ab}

Calculate the area of the square cutout.

Area_{cutout} = (x^3 - 2)^2 = x^6 - 4x^3 + 4

Squaring the binomial yields

(x^3)^2 - 2(2)(x^3) + 2^2

, which simplifies to

x^6 - 4x^3 + 4

Subtract the area of the cutout from the total area of the metal sheet.

Area_{remaining} = 9x^6 - (x^6 - 4x^3 + 4) = 8x^6 + 4x^3 - 4

Distributing the negative sign through the parentheses yields

9x^6 - x^6 + 4x^3 - 4

. Combining the like terms

9x^6

and

-x^6

gives the final simplified expression.

Key Concept

Polynomial subtraction and binomial expansion

Estimated Time:1m 30s

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