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:

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