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.
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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:
\[ (x^3 + x^2 - 14x - 24) - (x^3 - 4x^2) = 5x^2 - 14x - 24 \]
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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:
\[ (5x^2 - 14x - 24) - (5x^2 - 20x) = 6x - 24 \]
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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 \( (x + 2)(x + 3) \).