Question

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 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 \( x + 3 \).

1. **Divide the leading term:**
- First, divide \( 2x^3 \) by \( x \) to get \( 2x^2 \).

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

3. **Repeat the process:**
- Now divide \( -x^2 \) by \( x \) to get \( -x \).
- Multiply \( -x \) by \( x + 3 \):
\[
-x(x + 3) = -x^2 - 3x
\]
- Subtract:
\[
(-x^2 + 3x + 18) - (-x^2 - 3x) = 3x + 3x + 18 = 6x + 18
\]

4. **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:

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