To find the cross-sectional area of the rectangular solid shipping container, we need to divide the volume polynomial by the height polynomial. The volume \( V \) of the container is given by:
\[ V = 24x^3 + 6x^2 - 9x \]
The height \( h \) of the container is:
\[ h = 3x \]
To find the cross-sectional area \( A \), we divide the volume \( V \) by the height \( h \):
\[ A = \frac{V}{h} = \frac{24x^3 + 6x^2 - 9x}{3x} \]
We can perform polynomial long division to simplify this expression.
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Divide the leading term of the numerator by the leading term of the denominator:
\[ \frac{24x^3}{3x} = 8x^2 \]
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Multiply the entire denominator by this result and subtract it from the numerator:
\[ (24x^3 + 6x^2 - 9x) - (8x^2 \cdot 3x) = 24x^3 + 6x^2 - 9x - (24x^3) = 6x^2 - 9x \]
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Now divide the leading term of the remaining polynomial by the leading term of the denominator again:
\[ \frac{6x^2}{3x} = 2x \]
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Multiply the entire denominator by this result and subtract again:
\[ (6x^2 - 9x) - (2x \cdot 3x) = 6x^2 - 9x - (6x^2) = -9x \]
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Now divide the leading term of the remaining polynomial by the leading term of the denominator again:
\[ \frac{-9x}{3x} = -3 \]
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Multiply the entire denominator by this result and subtract again:
\[ (-9x) - (-3 \cdot 3x) = -9x + 9x = 0 \]
Now we combine all the terms we've obtained:
\[ A = 8x^2 + 2x - 3 \]
Therefore, the cross-sectional area of the container is:
\[ \boxed{8x^2 + 2x - 3} \]
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