To find the container's cross-sectional area, we need to divide the volume of the container by its height.
The volume of the container is given by: \[ V = 24x^3 + 6x^2 - 9x \] The height of the container is given by: \[ h = 3x \]
To find the cross-sectional area \(A\), we perform the polynomial division of the volume \(V\) by the height \(h\): \[ A = \frac{V}{h} = \frac{24x^3 + 6x^2 - 9x}{3x} \]
Now we will perform the polynomial division:
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Divide the leading term of the numerator \(24x^3\) by the leading term of the denominator \(3x\): \[ \frac{24x^3}{3x} = 8x^2 \]
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Multiply the entire divisor \(3x\) by \(8x^2\): \[ 8x^2 \cdot 3x = 24x^3 \]
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Subtract this from the original polynomial: \[ (24x^3 + 6x^2 - 9x) - 24x^3 = 6x^2 - 9x \]
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Now divide the leading term \(6x^2\) by the leading term of the denominator \(3x\): \[ \frac{6x^2}{3x} = 2x \]
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Multiply the entire divisor \(3x\) by \(2x\): \[ 2x \cdot 3x = 6x^2 \]
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Subtract this from the current polynomial: \[ (6x^2 - 9x) - 6x^2 = -9x \]
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Now divide the leading term \(-9x\) by the leading term of the denominator \(3x\): \[ \frac{-9x}{3x} = -3 \]
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Multiply the entire divisor \(3x\) by \(-3\): \[ -3 \cdot 3x = -9x \]
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Subtract this from the current polynomial: \[ (-9x) - (-9x) = 0 \]
At this point, we have finished the polynomial division, and we obtain: \[ A = 8x^2 + 2x - 3 \]
Thus, the container’s cross-sectional area is: \[ \boxed{8x^2 + 2x - 3} \]