To find the length of the rectangle, we can use the formula for the area of a rectangle, which is given by:
\[ \text{Area} = \text{Length} \times \text{Width} \]
In this case, the area is given as \( 43m^3 + 16m^2 - 23m - 15 \) ft², and the width is given as \( \left(m + \frac{1}{2}\right) \) ft.
We can find the length (L) by rearranging the area formula:
\[ L = \frac{\text{Area}}{\text{Width}} \]
Substituting the area and the width into this formula, we have:
\[ L = \frac{43m^3 + 16m^2 - 23m - 15}{m + \frac{1}{2}} \]
To perform the division, we can simplify \( m + \frac{1}{2} \) to a single fraction:
\[ L = \frac{43m^3 + 16m^2 - 23m - 15}{\frac{2m + 1}{2}} = \frac{2(43m^3 + 16m^2 - 23m - 15)}{2m + 1} \]
Now we need to divide \( 43m^3 + 16m^2 - 23m - 15 \) by \( 2m + 1 \).
We can perform polynomial long division:
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Divide the leading term of the numerator by the leading term of the divisor: \[ \frac{43m^3}{2m} = \frac{43}{2}m^2 \]
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Multiply \( \frac{43}{2}m^2 \) by \( 2m + 1 \): \[ \frac{43}{2}m^2(2m + 1) = 43m^3 + \frac{43}{2}m^2 \]
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Subtract this from the original polynomial: \[ (43m^3 + 16m^2 - 23m - 15) - \left(43m^3 + \frac{43}{2}m^2\right) = \left(16m^2 - \frac{43}{2}m^2\right) - 23m - 15 \]
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Combine like terms: \[ 16m^2 - \frac{43}{2}m^2 = \frac{32m^2 - 43m^2}{2} = \frac{-11m^2}{2} \] So, we have: \[ -\frac{11}{2}m^2 - 23m - 15 \]
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Repeat the process by taking the new leading term: \[ \frac{-\frac{11}{2}m^2}{2m} = -\frac{11}{4}m \]
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Multiply: \[ -\frac{11}{4}m(2m + 1) = -\frac{11}{2}m^2 - \frac{11}{4}m \]
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Subtract again: \[ \left(-\frac{11}{2}m^2 - 23m - 15\right) - \left(-\frac{11}{2}m^2 - \frac{11}{4}m\right) = \left(-23 + \frac{11}{4}\right)m - 15 = -\frac{69}{4}m - 15 \]
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Continue the process with: \[ \frac{-\frac{69}{4}m}{2m} = -\frac{69}{8} \]
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Multiply: \[ -\frac{69}{8}(2m + 1) = -\frac{69}{4}m - \frac{69}{8} \]
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Finally, subtract: \[ -\frac{69}{4}m - 15 + \frac{69}{4}m + \frac{69}{8} = -15 + \frac{69}{8} \]
Converting -15 to eighths: \[ -15 = -\frac{120}{8} \] So the remainder becomes: \[ -\frac{120}{8} + \frac{69}{8} = -\frac{51}{8} \]
Putting this all together, we find: \[ L = \frac{43}{2}m^2 - \frac{11}{4}m - \frac{69}{8} \]
Thus, the final expression for the length \(L\) of the rectangle, given the width, is:
\[ L = \frac{2(43m^3 + 16m^2 - 23m - 15)}{2m + 1} = 43m^2 - 11m - \frac{69}{8} \]
This polynomial represents the length of the rectangle in terms of \(m\).