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 2x^3+5x^2+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?

Answer 1

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

1. **Divide the leading terms**:
\( 2x^3 \div x = 2x^2 \).

2. **Multiply** \( 2x^2 \) by \( x + 3 \):
\[
2x^2 \cdot (x + 3) = 2x^3 + 6x^2
\]

3. **Subtract** this from the original polynomial:
\[
(2x^3 + 5x^2 + 3x + 18) - (2x^3 + 6x^2) = 5x^2 - 6x^2 + 3x + 18 = -x^2 + 3x + 18
\]

4. **Divide the leading terms**:
\( -x^2 \div x = -x \).

5. **Multiply** \( -x \) by \( x + 3 \):
\[
-x \cdot (x + 3) = -x^2 - 3x
\]

6. **Subtract**:
\[
(-x^2 + 3x + 18) - (-x^2 - 3x) = 3x + 3x + 18 = 6x + 18
\]

7. **Divide the leading terms**:
\( 6x \div x = 6 \).

8. **Multiply** \( 6 \) by \( x + 3 \):
\[
6 \cdot (x + 3) = 6x + 18
\]

9. **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:

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