To find the area of the base of the cylinder, we can use the formula for the volume of a cylinder, which is given by:
\[ \text{Volume} = \text{Base Area} \times \text{Height} \]
In this problem, we are given the volume \( V = 2x^3 + 5x^2 + 3x + 18 \) and the height \( h = x + 3 \). We need to find the area of the base (let's call it \( A \)). We can rearrange the formula to solve for the area of the base:
\[ A = \frac{\text{Volume}}{\text{Height}} \]
To find \( A \), we will divide the polynomial for volume \( V \) by the polynomial for height \( h \).
Step 1: Divide the polynomials
We will divide the polynomial \( 2x^3 + 5x^2 + 3x + 18 \) by \( x + 3 \).
Polynomial Long Division
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Divide the leading terms: \( 2x^3 \div x = 2x^2 \).
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Multiply \( 2x^2 \) by \( x + 3 \): \[ 2x^2 \cdot (x + 3) = 2x^3 + 6x^2 \]
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Subtract this from the original polynomial: \[ (2x^3 + 5x^2 + 3x + 18) - (2x^3 + 6x^2) = 5x^2 - 6x^2 + 3x + 18 = -x^2 + 3x + 18 \]
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Divide the leading terms: \( -x^2 \div x = -x \).
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Multiply \( -x \) by \( x + 3 \): \[ -x \cdot (x + 3) = -x^2 - 3x \]
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Subtract: \[ (-x^2 + 3x + 18) - (-x^2 - 3x) = 3x + 3x + 18 = 6x + 18 \]
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Divide the leading terms: \( 6x \div x = 6 \).
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Multiply \( 6 \) by \( x + 3 \): \[ 6 \cdot (x + 3) = 6x + 18 \]
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Subtract: \[ (6x + 18) - (6x + 18) = 0 \]
Conclusion
After the division, we have no remainder. The result of the polynomial long division is:
\[ A = 2x^2 - x + 6 \]
Thus, the expression that represents the area of the base of the cylinder is:
\[ \boxed{2x^2 - x + 6} \]