To find the expression for the area of the base of the cylinder, we can use the formula for the volume of a cylinder:
\[ \text{Volume} = \text{Base Area} \times \text{Height} \]
In this case, we know the following:
- The volume of the cylinder is given by the polynomial \( V = 2x^3 + 5x^2 + 3x + 18 \).
- The height of the cylinder is given by the polynomial \( h = x + 3 \).
We can express the area of the base, \( A \), as follows:
\[ A = \frac{V}{h} = \frac{2x^3 + 5x^2 + 3x + 18}{x + 3} \]
Now, we need to perform polynomial long division to divide \( 2x^3 + 5x^2 + 3x + 18 \) by \( x + 3 \).
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Divide the leading term:
- First, divide \( 2x^3 \) by \( x \) to get \( 2x^2 \).
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Multiply and subtract:
- Multiply \( 2x^2 \) by \( x + 3 \): \[ 2x^2(x + 3) = 2x^3 + 6x^2 \]
- 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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Repeat the process:
- Now divide \( -x^2 \) by \( x \) to get \( -x \).
- Multiply \( -x \) by \( x + 3 \): \[ -x(x + 3) = -x^2 - 3x \]
- Subtract: \[ (-x^2 + 3x + 18) - (-x^2 - 3x) = 3x + 3x + 18 = 6x + 18 \]
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Repeat the process again:
- Now divide \( 6x \) by \( x \) to get \( 6 \).
- Multiply \( 6 \) by \( x + 3 \): \[ 6(x + 3) = 6x + 18 \]
- Subtract: \[ (6x + 18) - (6x + 18) = 0 \]
Since we have a remainder of 0, the division is exact.
Thus, the area of the base of the cylinder is represented by the expression:
\[ A = 2x^2 - x + 6 \]
Therefore, the expression representing the area of the base of the cylinder is:
\[ \boxed{2x^2 - x + 6} \]