To determine the width of the rectangular field, we need to divide the area polynomial \( x^3 + x^2 - 14x - 24 \) by the length polynomial \( x - 4 \).
We can perform polynomial long division:
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Divide the leading term of the area \( x^3 \) by the leading term of the length \( x \) to get \( x^2 \).
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Multiply \( x^2 \) by \( x - 4 \) to get \( x^3 - 4x^2 \).
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Subtract \( (x^3 - 4x^2) \) from \( (x^3 + x^2 - 14x - 24) \): \[ (x^3 + x^2 - 14x - 24) - (x^3 - 4x^2) = 5x^2 - 14x - 24 \]
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Next, divide the leading term of the new polynomial \( 5x^2 \) by \( x \) to get \( 5x \).
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Multiply \( 5x \) by \( x - 4 \) to get \( 5x^2 - 20x \).
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Subtract \( (5x^2 - 20x) \) from \( (5x^2 - 14x - 24) \): \[ (5x^2 - 14x - 24) - (5x^2 - 20x) = 6x - 24 \]
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Now, divide the leading term of the new polynomial \( 6x \) by \( x \) to get \( 6 \).
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Multiply \( 6 \) by \( x - 4 \) to get \( 6x - 24 \).
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Subtract \( (6x - 24) \) from \( (6x - 24) \): \[ (6x - 24) - (6x - 24) = 0 \]
Now that there is no remainder, we have completed the long division, and we can express the area polynomial as:
\[ \text{Area} = (\text{Length}) \times (\text{Width}) \implies x^3 + x^2 - 14x - 24 = (x - 4)(x^2 + 5x + 6) \]
Thus, the expression that represents the width of the rectangular field is:
\[ x^2 + 5x + 6 \]
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