Question

The area of a rectangular field is represented by the polynomial x3+x2−14x−24
. The length of the field is represented by the polynomial x−4
. Determine the expression that represents the width of the field.(1 point)
$$

Answer 1

To find the width of the rectangular field, we can use polynomial long division to divide the area polynomial \( x^3 + x^2 - 14x - 24 \) by the length polynomial \( x - 4 \).

We start with the long division:

1. Divide the leading term of the area polynomial \( x^3 \) by the leading term of the length polynomial \( x \) to get \( x^2 \).
2. Multiply \( x^2 \) by \( x - 4 \) to get \( x^3 - 4x^2 \).
3. Subtract this from the original polynomial:
\[
(x^3 + x^2 - 14x - 24) - (x^3 - 4x^2) = 5x^2 - 14x - 24
\]

4. Now repeat the process with \( 5x^2 \):
- Divide \( 5x^2 \) by \( x \) to get \( 5x \).
- Multiply \( 5x \) by \( x - 4 \) to get \( 5x^2 - 20x \).
- Subtract:
\[
(5x^2 - 14x - 24) - (5x^2 - 20x) = 6x - 24
\]

5. Finally, take \( 6x \):
- Divide \( 6x \) by \( x \) to get \( 6 \).
- Multiply \( 6 \) by \( x - 4 \) to get \( 6x - 24 \).
- Subtract:
\[
(6x - 24) - (6x - 24) = 0
\]

The result of the division is:

\[
x^2 + 5x + 6
\]

Therefore, the expression that represents the width of the field is:

\[
\boxed{x^2 + 5x + 6}
\]