To find the length of the rectangle given the area and the width, we can use the formula for the area of a rectangle:
\[ \text{Area} = \text{Length} \times \text{Width} \]
Given that the area of the rectangle is \(4m^3 + 16m^2 - 23m - 15\) ft² and the width is \(\frac{m + 1}{2}\) ft, we can rearrange the formula to find the length:
\[ \text{Length} = \frac{\text{Area}}{\text{Width}} \]
Substituting in the given area and width:
\[ \text{Length} = \frac{4m^3 + 16m^2 - 23m - 15}{\frac{m + 1}{2}} \]
To simplify this expression, we can multiply the numerator by the reciprocal of the width:
\[ \text{Length} = \left(4m^3 + 16m^2 - 23m - 15\right) \times \frac{2}{m + 1} \]
Now we can simplify this by performing polynomial long division or factoring, if possible, to divide \(4m^3 + 16m^2 - 23m - 15\) by \(m + 1\).
Let's perform polynomial long division:
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Divide the leading term of the numerator \(4m^3\) by the leading term of the denominator \(m\): \[ 4m^2 \]
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Multiply \(4m^2\) by \(m + 1\): \[ 4m^3 + 4m^2 \]
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Subtract this from the original polynomial: \[ (4m^3 + 16m^2 - 23m - 15) - (4m^3 + 4m^2) = 12m^2 - 23m - 15 \]
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Divide the leading term \(12m^2\) by \(m\): \[ 12m \]
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Multiply \(12m\) by \(m + 1\): \[ 12m^2 + 12m \]
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Subtract this from the previous remainder: \[ (12m^2 - 23m - 15) - (12m^2 + 12m) = -35m - 15 \]
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Divide the leading term \(-35m\) by \(m\): \[ -35 \]
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Multiply \(-35\) by \(m + 1\): \[ -35m - 35 \]
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Subtract this: \[ (-35m - 15) - (-35m - 35) = 20 \]
Thus, the division gives us:
\[ 4m^2 + 12m - 35 + \frac{20}{m + 1} \]
Now substituting back to find the length:
\[ \text{Length} = 2(4m^2 + 12m - 35) + \frac{40}{m + 1} \]
Simplifying further:
\[ \text{Length} = 8m^2 + 24m - 70 + \frac{40}{m + 1} \]
The final answer for the length of the rectangle is:
\[ \text{Length} = 8m^2 + 24m - 70 + \frac{40}{m + 1} \text{ ft} \]
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