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. 21x^2−x−103x+2

Answer 1

To find the width of the rectangle, we need to divide the area by the length.

The area of the rectangle is given as \(21x^2 - x - 10\), and the length is given as \(3x + 2\). Therefore, we need to perform the division:

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

We'll perform polynomial long division to divide \(21x^2 - x - 10\) by \(3x + 2\).

1. **Divide the leading term:**
- The leading term of the numerator is \(21x^2\) and the leading term of the denominator is \(3x\).
- Dividing gives:
\[
\frac{21x^2}{3x} = 7x.
\]

2. **Multiply and subtract:**
- Multiply \(7x\) by \(3x + 2\):
\[
7x(3x + 2) = 21x^2 + 14x.
\]
- Now, subtract this from the original polynomial:
\[
(21x^2 - x - 10) - (21x^2 + 14x) = -x - 14x - 10 = -15x - 10.
\]

3. **Repeat the process:**
- Now, divide the new leading term \(-15x\) by \(3x\):
\[
\frac{-15x}{3x} = -5.
\]
- Multiply \(-5\) by \(3x + 2\):
\[
-5(3x + 2) = -15x - 10.
\]
- Now, subtract this from \(-15x - 10\):
\[
(-15x - 10) - (-15x - 10) = 0.
\]

Since the remainder is \(0\), we find that:

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

Thus, the width of the rectangle is:

\[
\text{Width} = 7x - 5.
\]