To simplify the expression \((43m^3 + 16m^2 - 23m - 15) \div \left(m + \frac{1}{2}\right)\), we can multiply \(m + \frac{1}{2}\) by 2 to eliminate the fraction. This gives us:
\[ (43m^3 + 16m^2 - 23m - 15) \div (m + \frac{1}{2}) = (43m^3 + 16m^2 - 23m - 15) \div \left(\frac{2m + 1}{2}\right) \]
This is the same as multiplying by the reciprocal:
\[ (43m^3 + 16m^2 - 23m - 15) \cdot \left(\frac{2}{2m + 1}\right) \]
Now let's perform polynomial long division on \(43m^3 + 16m^2 - 23m - 15\) divided by \(2m + 1\).
Step 1: Polynomial Long Division
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Divide the leading term of the dividend \(43m^3\) by the leading term of the divisor \(2m\):
\[ \frac{43m^3}{2m} = \frac{43}{2} m^2 \]
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Multiply \(\frac{43}{2} m^2\) by the entire divisor \(2m + 1\):
\[ \frac{43}{2} m^2 (2m + 1) = 43m^3 + \frac{43}{2} m^2 \]
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Subtract this result from the original polynomial:
\[ (43m^3 + 16m^2 - 23m - 15) - (43m^3 + \frac{43}{2} m^2) = 16m^2 - \frac{43}{2} m^2 - 23m - 15 \]
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Combine like terms:
\[ 16m^2 - \frac{43}{2} m^2 = \left(\frac{32}{2} m^2 - \frac{43}{2} m^2\right) = -\frac{11}{2} m^2 \]
So now we have:
\[ -\frac{11}{2} m^2 - 23m - 15 \]
Step 2: Repeat the process
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Divide the leading term \(-\frac{11}{2} m^2\) by \(2m\):
\[ \frac{-\frac{11}{2} m^2}{2m} = -\frac{11}{4} m \]
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Multiply \(-\frac{11}{4} m\) by \(2m + 1\):
\[ -\frac{11}{4} m (2m + 1) = -\frac{11}{2} m^2 - \frac{11}{4} m \]
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Subtract:
\[ (-\frac{11}{2} m^2 - 23m - 15) - (-\frac{11}{2} m^2 - \frac{11}{4} m) = -23m + \frac{11}{4} m - 15 \]
- Combine like terms:
\[ -23m + \frac{11}{4} m = \left(-\frac{92}{4} m + \frac{11}{4} m\right) = -\frac{81}{4} m \]
Step 3: Final Division
Now combine it with \(-15\):
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Divide \(-\frac{81}{4} m\) by \(2m\):
\[ \frac{-\frac{81}{4} m}{2m} = -\frac{81}{8} \]
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Multiply by \(2m + 1\):
\[ -\frac{81}{8} \cdot (2m + 1) = -\frac{81}{4} m - \frac{81}{8} \]
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Subtract from the earlier remainder:
\[ -\frac{81}{4} m - 15 + \left(-\frac{81}{4} m + \frac{81}{8}\right) \]
Conclusion
At this final step, the remainder persists as a constant. After simplification, we could express the complete solution as:
\[ \frac{43}{2} m^2 - \frac{11}{4} m - \frac{81}{8} \]
Now the answer is:
\[ \frac{43}{2} m^2 - \frac{11}{4} m - \frac{81}{8} + R \]
Full Expression
Thus, the simplified result of:
\[ (43m^3 + 16m^2 - 23m - 15) \div \left(m + \frac{1}{2}\right) = 21.5m^2 - 2.75 m - 10.125 \]
Please adjust as necessary according to additional context!