Question

The length of a rectangle is 3x+2 . Its area is 21x2−x−10 . To find an expression that represents the width of the rectangle, perform the following operation(s) and simplify completely. 21x2−x−103x+2 (1 point)

Answer 1

To find the width of the rectangle, we need to divide the area \(21x^2 - x - 10\) by the length \(3x + 2\).

Let's perform the polynomial long division:

1. **Divide the leading term of the numerator by the leading term of the denominator**:
\[
\frac{21x^2}{3x} = 7x
\]
So, we will multiply the entire length \(3x + 2\) by \(7x\):
\[
7x(3x + 2) = 21x^2 + 14x
\]

2. **Subtract this result from the area**:
\[
(21x^2 - x - 10) - (21x^2 + 14x) = -x - 10 - 14x = -15x - 10
\]

3. **Now, divide the leading term of the new expression by the leading term of the denominator**:
\[
\frac{-15x}{3x} = -5
\]
Now multiply the entire length \(3x + 2\) by \(-5\):
\[
-5(3x + 2) = -15x - 10
\]

4. **Subtract this result from the previous remainder**:
\[
(-15x - 10) - (-15x - 10) = 0
\]

Since the remainder is 0, the division is exact.

Therefore, the width of the rectangle is:
\[
\text{Width} = 7x - 5
\]

Thus, the expression representing the width of the rectangle is:
\[
\boxed{7x - 5}
\]