To find the expression that represents the width of the field, we need to divide the polynomial representing the area of the rectangular field by the polynomial representing the length of the field.
Given:
- Area \( A = x^3 + x^2 - 14x - 24 \)
- Length \( L = x - 4 \)
We will perform polynomial long division of \( A \) by \( L \).
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Divide the first term of the area by the first term of the length:
\[ \frac{x^3}{x} = x^2 \]
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Multiply \( x^2 \) by \( L \):
\[ x^2(x - 4) = x^3 - 4x^2 \]
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Subtract this from the original polynomial \( A \):
\[ (x^3 + x^2 - 14x - 24) - (x^3 - 4x^2) = (x^2 + 4x^2) - 14x - 24 = 5x^2 - 14x - 24 \]
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Now, divide the first term of the new polynomial by the first term of the length:
\[ \frac{5x^2}{x} = 5x \]
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Multiply \( 5x \) by \( L \):
\[ 5x(x - 4) = 5x^2 - 20x \]
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Subtract this from the current polynomial:
\[ (5x^2 - 14x - 24) - (5x^2 - 20x) = (-14x + 20x) - 24 = 6x - 24 \]
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Now divide the first term of the result by the first term of the length:
\[ \frac{6x}{x} = 6 \]
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Multiply \( 6 \) by \( L \):
\[ 6(x - 4) = 6x - 24 \]
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Subtract this from the current polynomial:
\[ (6x - 24) - (6x - 24) = 0 \]
Thus, the polynomial \( A = x^3 + x^2 - 14x - 24 \) can be factored as:
\[ A = (x - 4)(x^2 + 5x + 6) \]
So, the expression that represents the width of the field is:
\[ \text{Width} = x^2 + 5x + 6 \]
Therefore, the expression that represents the width of the field is:
\[ \boxed{x^2 + 5x + 6} \]
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