Question

The area of a rectangular carpet is 10x2−3x−27
. The length of the carpet is 5x−9
yards. Complete the statement to describe the width of the rectangle.

The width of the rectangle is (Select)
(Select)
2x + 3
2x - 3
5x - 3
5x + 3

Answer 1

To find the width of the rectangular carpet, we need to use the formula for the area of a rectangle, which is:

\[
\text{Area} = \text{Length} \times \text{Width}
\]

We know from the problem that the area is \(10x^2 - 3x - 27\) and the length is \(5x - 9\). We can express the width by rearranging the area formula:

\[
\text{Width} = \frac{\text{Area}}{\text{Length}}
\]

Substituting the values we have:

\[
\text{Width} = \frac{10x^2 - 3x - 27}{5x - 9}
\]

Next, we need to simplify this expression. We'll perform polynomial long division or synthetic division.

1. First, divide \(10x^2\) by \(5x\) to get \(2x\).
2. Multiply \(2x\) by \(5x - 9\) to get \(10x^2 - 18x\).
3. Subtract this from \(10x^2 - 3x - 27\):

\[
(10x^2 - 3x - 27) - (10x^2 - 18x) = (18x - 3x) - 27 = 15x - 27
\]

4. Now, divide \(15x\) by \(5x\) to get \(3\).
5. Multiply \(3\) by \(5x - 9\) to get \(15x - 27\).
6. Subtract this from \(15x - 27\):

\[
(15x - 27) - (15x - 27) = 0
\]

Therefore, the division results in:

\[
\text{Width} = 2x + 3
\]

Thus, we can complete the statement:

The width of the rectangle is **2x + 3**.