To find the area of the base of the cylinder, you can use the relationship between the volume of a cylinder, the area of the base, and the height:
\[ \text{Volume} = \text{Area of Base} \times \text{Height} \]
In this problem:
- The volume of the cylinder is given by the polynomial \(2x^3 + 5x^2 + 3x + 18\).
- The height of the cylinder is given by the polynomial \(x + 3\).
We need to find the area of the base, which can be represented as:
\[ \text{Area of Base} = \frac{\text{Volume}}{\text{Height}} \]
To find the area of the base, we will perform polynomial long division to divide the volume polynomial by the height polynomial.
Step 1: Divide the polynomials
We need to divide \(2x^3 + 5x^2 + 3x + 18\) by \(x + 3\).
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The first term is \(2x^3\). Divide by \(x\) to get \(2x^2\).
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Multiply \(2x^2\) by \(x + 3\) to get \(2x^3 + 6x^2\).
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Subtract \(2x^3 + 6x^2\) from \(2x^3 + 5x^2 + 3x + 18\):
\[ (2x^3 + 5x^2 + 3x + 18) - (2x^3 + 6x^2) = (5x^2 - 6x^2) + 3x + 18 = -x^2 + 3x + 18 \]
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The next term is \(-x^2\). Divide by \(x\) to get \(-x\).
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Multiply \(-x\) by \(x + 3\) to get \(-x^2 - 3x\).
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Subtract this from \(-x^2 + 3x + 18\):
\[ (-x^2 + 3x + 18) - (-x^2 - 3x) = 3x + 3x + 18 = 6x + 18 \]
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The next term is \(6x\). Divide by \(x\) to get \(6\).
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Multiply \(6\ by (x + 3\) to get \(6x + 18\).
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Subtract this from \(6x + 18\):
\[ (6x + 18) - (6x + 18) = 0 \]
Final Result
No remainder is left, so the division gives:
\[ \text{Area of Base} = 2x^2 - x + 6 \]
Therefore, the expression that represents the area of the base of the cylinder is:
\[ \boxed{2x^2 - x + 6} \]