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.
- First, divide \(10x^2\) by \(5x\) to get \(2x\).
- Multiply \(2x\) by \(5x - 9\) to get \(10x^2 - 18x\).
- Subtract this from \(10x^2 - 3x - 27\):
\[ (10x^2 - 3x - 27) - (10x^2 - 18x) = (18x - 3x) - 27 = 15x - 27 \]
- Now, divide \(15x\) by \(5x\) to get \(3\).
- Multiply \(3\) by \(5x - 9\) to get \(15x - 27\).
- 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.