To find the missing term in the product \((−9m^6+12m^5)(m^2+2m+1)=−9m^8−6m^7+?+12m^5\), we can break down the multiplication step by step, using the distributive property (also known as the FOIL method for binomials):
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Expand the left side:
We will distribute \((-9m^6)\) and \((12m^5)\) to each term in \((m^2 + 2m + 1)\).
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First term: \(-9m^6 \cdot m^2\): \[ -9m^8 \]
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Second term: \(-9m^6 \cdot 2m\): \[ -18m^7 \]
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Third term: \(-9m^6 \cdot 1\): \[ -9m^6 \]
Now for the second polynomial term:
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Fourth term: \(12m^5 \cdot m^2\): \[ 12m^7 \]
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Fifth term: \(12m^5 \cdot 2m\): \[ 24m^6 \]
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Sixth term: \(12m^5 \cdot 1\): \[ 12m^5 \]
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Combine all the terms:
Now we can combine all the results:
- Collect the \(m^8\) term: \[ -9m^8 \]
- Collect the \(m^7\) terms: \[ -18m^7 + 12m^7 = -6m^7 \]
- Collect the \(m^6\) terms: \[ -9m^6 + 24m^6 = 15m^6 \]
- The constant term is: \[ 12m^5 \]
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Final expansion:
Thus, the entire expansion is: \[ -9m^8 - 6m^7 + 15m^6 + 12m^5 \]
Now, looking at the product we are supposed to equate this to, we have: \[ -9m^8 - 6m^7 + ? + 12m^5 \] By comparing the coefficients of \(m^6\), we find: \[ ? = 15m^6 \]
Hence, the missing term is \(15m^6\).
Conclusion
The answer is: 15m^6