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 determine the width of the rectangular field represented by the polynomial area \( A(x) = x^3 + x^2 - 14x - 24 \) and length represented by \( L(x) = x - 4 \), we can find the width by dividing the area by the length:

\[
W(x) = \frac{A(x)}{L(x)} = \frac{x^3 + x^2 - 14x - 24}{x - 4}
\]

To perform this polynomial division, we can use synthetic division or long division. Here, I will use polynomial long division.

1. **Set up the division**:

Divide \( x^3 \) by \( x \) to get \( x^2 \).

Multiply \( x^2 \) by \( x - 4 \) to get \( x^3 - 4x^2 \).

Subtract this from the original polynomial:

\[
(x^3 + x^2 - 14x - 24) - (x^3 - 4x^2) = 5x^2 - 14x - 24
\]

2. **Continue dividing**:

Now divide \( 5x^2 \) by \( x \) to get \( 5x \).

Multiply \( 5x \) by \( x - 4 \) to get \( 5x^2 - 20x \).

Subtract this from the new polynomial:

\[
(5x^2 - 14x - 24) - (5x^2 - 20x) = 6x - 24
\]

3. **Final step**:

Now divide \( 6x \) by \( x \) to get \( 6 \).

Multiply \( 6 \) by \( x - 4 \) to get \( 6x - 24 \).

Subtract this from the polynomial:

\[
(6x - 24) - (6x - 24) = 0
\]

Now, putting it all together, we have:

\[
W(x) = x^2 + 5x + 6
\]

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

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

This expression can also be factored if needed. It factors to \( (x + 2)(x + 3) \).