Question

Construct a table to find the missing term in the product (−9m6+12m5)(m2+2m+1)=−9m8−6m7+?+12m5 .(1 point)

Answer 1

To find the missing term in the product \((−9m^6+12m^5)(m^2+2m+1)=−9m^8−6m^7+?+12m^5\), we first expand the left-hand side.

Given:
\[
A = -9m^6 + 12m^5
\]
\[
B = m^2 + 2m + 1
\]

Now, let's perform the multiplication \(A \cdot B\):

\[
A \cdot B = (-9m^6)(m^2) + (-9m^6)(2m) + (-9m^6)(1) + (12m^5)(m^2) + (12m^5)(2m) + (12m^5)(1)
\]

Calculating each term:
1. \((-9m^6)(m^2) = -9m^8\)
2. \((-9m^6)(2m) = -18m^7\)
3. \((-9m^6)(1) = -9m^6\)
4. \((12m^5)(m^2) = 12m^7\)
5. \((12m^5)(2m) = 24m^6\)
6. \((12m^5)(1) = 12m^5\)

Now we will combine like terms:
- The \(m^8\) term is \(-9m^8\).
- The \(m^7\) terms are \(-18m^7 + 12m^7 = -6m^7\).
- The \(m^6\) terms are \(-9m^6 + 24m^6 = 15m^6\).
- The \(m^5\) term is \(12m^5\).

Thus, the overall product is:
\[
-9m^8 - 6m^7 + 15m^6 + 12m^5
\]

Now we compare this to the given expression:
\[
−9m^8−6m^7+?+12m^5
\]

We can see that the missing term, which we have calculated as \(15m^6\), should be represented as \(?\).

Thus, the missing term is:
\[
\boxed{15m^6}
\]