Question

Explain which operation to use to solve the following problem. Then, solve the problem.The volume of a cylinder is represented by the polynomial 2x3+5x2+3x+18 . The height of the cylinder is represented by the polynomial x+3 . What expression represents the area of the base of the cylinder?(2 points)

Answer 1

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 \(x + 3\).

1. The first term is \(2x^3\). Divide by \(x\) to get \(2x^2\).
2. Multiply \(2x^2\) by \(x + 3\) to get \(2x^3 + 6x^2\).
3. Subtract \(2x^3 + 6x^2\) from \(2x^3 + 5x^2 + 3x + 18\):

\[
(2x^3 + 5x^2 + 3x + 18) - (2x^3 + 6x^2) = (5x^2 - 6x^2) + 3x + 18 = -x^2 + 3x + 18
\]

4. The next term is \(-x^2\). Divide by \(x\) to get \(-x\).
5. Multiply \(-x\) by \(x + 3\) to get \(-x^2 - 3x\).
6. Subtract this from \(-x^2 + 3x + 18\):

\[
(-x^2 + 3x + 18) - (-x^2 - 3x) = 3x + 3x + 18 = 6x + 18
\]

7. The next term is \(6x\). Divide by \(x\) to get \(6\).
8. Multiply \(6\ by \(x + 3\) to get \(6x + 18\).
9. 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:

\[
\boxed{2x^2 - x + 6}
\]