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
A) 2x + 3
B) 2x 3
C) 5x 3
D) 5x + 3

Answer 1

To find the width of the rectangle, we can use the formula for the area of a rectangle, which is:

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

In this case, the area of the carpet is given as \( 10x^2 - 3x - 27 \) and the length is given as \( 5x - 9 \). We can represent the width as \( W \).

Setting up the equation:

\[
10x^2 - 3x - 27 = (5x - 9) \times W
\]

We can find the width \( W \) by dividing the area by the length:

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

Now, we will perform polynomial long division.

1. Divide the leading term of the numerator \( 10x^2 \) by the leading term of the denominator \( 5x \), which gives \( 2x \).
2. Multiply \( 2x \) by \( 5x - 9 \):
\[
2x(5x - 9) = 10x^2 - 18x
\]
3. Subtract this from the original polynomial:
\[
(10x^2 - 3x - 27) - (10x^2 - 18x) = 15x - 27
\]
4. Now, divide the leading term of the new polynomial \( 15x \) by the leading term of the denominator \( 5x \), giving \( 3 \).
5. Multiply \( 3 \) by \( 5x - 9 \):
\[
3(5x - 9) = 15x - 27
\]
6. Subtract this:
\[
(15x - 27) - (15x - 27) = 0
\]

The division indicates that:

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

Thus, the width of the rectangle is:

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

Therefore, the correct answer is:

**A) 2x + 3**.