Question

Construct a table to find the missing term in the product (−9m6+12m5)(m2+2m+1)=−9m8−6m7+?+12m5 .(1 point)%0D%0AResponses%0D%0A%0D%0A15m6%0D%0A15 m superscript 6 baseline%0D%0A%0D%0A33m6%0D%0A33 m superscript 6 baseline%0D%0A%0D%0A−9m6%0D%0Anegative 9 m superscript 6 baseline%0D%0A%0D%0A3m6%0D%0A3 m superscript 6 baseline

Answer 1

To find the missing term in the product \( (-9m^6 + 12m^5)(m^2 + 2m + 1) \), we can use distribution (also known as the FOIL method for binomials). The expression can be expanded as follows:

1. **Distributing \(-9m^6\)**:
- \(-9m^6 \cdot m^2 = -9m^8\)
- \(-9m^6 \cdot 2m = -18m^7\)
- \(-9m^6 \cdot 1 = -9m^6\)

2. **Distributing \(12m^5\)**:
- \(12m^5 \cdot m^2 = 12m^7\)
- \(12m^5 \cdot 2m = 24m^6\)
- \(12m^5 \cdot 1 = 12m^5\)

Now, we can combine the terms we obtained from the distributions:

- From \(-9m^6\) we have:
- \(-9m^8\)
- \(-18m^7\)
- \(-9m^6\)

- From \(12m^5\) we have:
- \(12m^7\)
- \(24m^6\)
- \(12m^5\)

Combining these results gives the complete expansion:
\[
-9m^8 + (-18m^7 + 12m^7) + (-9m^6 + 24m^6) + 12m^5
\]

Let's simplify the terms:

- The \(m^8\) term: \(-9m^8\)
- The \(m^7\) term: \(-18m^7 + 12m^7 = -6m^7\)
- The \(m^6\) term: \(-9m^6 + 24m^6 = 15m^6\)
- The \(m^5\) term: \(12m^5\)

Thus, we can summarize the entire polynomial as:
\[
-9m^8 - 6m^7 + 15m^6 + 12m^5
\]

Now, looking back at your question:

The product gives us:
\[
(-9m^8 - 6m^7 + ? + 12m^5)
\]

We identified that the missing term is \(15m^6\).

Therefore, the correct response is:
**15m^6**.