Question

The length of a rectangle is 3x+2 . Its area is 21x^2−x−10 . To find an expression that represents the width of the rectangle, perform the following operation(s) and simplify completely. 21x2−x−10/3x+2 (1 point)

Answer 1

To find the width of the rectangle given its length and area, we can use the following relationship:

\[
\text{Width} = \frac{\text{Area}}{\text{Length}} = \frac{21x^2 - x - 10}{3x + 2}
\]

To find the expression for the width, we need to divide \(21x^2 - x - 10\) by \(3x + 2\). We will perform polynomial long division to simplify this expression.

1. **Set up the division**: Divide \(21x^2\) by \(3x\), which gives \(7x\).
2. **Multiply \(7x\) by \(3x + 2\)**:
\[
7x(3x + 2) = 21x^2 + 14x
\]
3. **Subtract this from the original polynomial**:
\[
(21x^2 - x - 10) - (21x^2 + 14x) = -x - 14x - 10 = -15x - 10
\]
4. **Now, divide \(-15x\) by \(3x\)**: This gives \(-5\).
5. **Multiply \(-5\) by \(3x + 2\)**:
\[
-5(3x + 2) = -15x - 10
\]
6. **Subtract this from the previous result**:
\[
(-15x - 10) - (-15x - 10) = 0
\]

Now we can conclude that \(21x^2 - x - 10\) is exactly divisible by \(3x + 2\), and the division gives us:

\[
\frac{21x^2 - x - 10}{3x + 2} = 7x - 5
\]

Thus, the expression that represents the width of the rectangle is:

\[
\boxed{7x - 5}
\]