To find the cross-sectional area of the rectangular solid shipping container given its volume \( V(x) = 24x^3 + 6x^2 - 9x \) and height \( h = 3x \), we will perform polynomial division to divide the volume by the height.
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Write down the expressions:
- Volume: \( V(x) = 24x^3 + 6x^2 - 9x \)
- Height: \( h(x) = 3x \)
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Perform polynomial division: We want to divide \( V(x) \) by \( h(x) \): \[ \frac{V(x)}{h(x)} = \frac{24x^3 + 6x^2 - 9x}{3x} \]
To perform the division, we can factor out \( 3x \):
- Divide the leading term of the numerator by the leading term of the denominator: \[ \frac{24x^3}{3x} = 8x^2 \]
- Multiply \( 8x^2 \) by \( 3x \): \[ 8x^2 \cdot 3x = 24x^3 \]
- Subtract this from the original polynomial: \[ (24x^3 + 6x^2 - 9x) - 24x^3 = 6x^2 - 9x \]
Now divide the next term:
- Divide the leading term of the modified numerator by the leading term of the denominator: \[ \frac{6x^2}{3x} = 2x \]
- Multiply \( 2x \) by \( 3x \): \[ 2x \cdot 3x = 6x^2 \]
- Subtract: \[ (6x^2 - 9x) - 6x^2 = -9x \]
Now divide the next term:
- Divide the leading term: \[ \frac{-9x}{3x} = -3 \]
- Multiply \( -3 \) by \( 3x \): \[ -3 \cdot 3x = -9x \]
- Subtract: \[ -9x - (-9x) = 0 \]
After the division, we've found: \[ \frac{V(x)}{h(x)} = 8x^2 + 2x - 3 \]
- Cross-sectional area: Therefore, the cross-sectional area \( A(x) \) of the container is given by: \[ A(x) = 8x^2 + 2x - 3 \]
Thus, the container's cross-sectional area is: \[ \boxed{8x^2 + 2x - 3} \text{ square inches.} \]
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