Question

(9m^6+12m^5)(m^2+2m+1)=-9m^8-6m^7+?+12m^5 construct a table to find the missing term in the product

Answer 1

To find the missing term in the expression \((9m^6+12m^5)(m^2+2m+1)\), we can use the distributive property and construct a table to visualize how the terms combine.

We begin by expanding the two polynomials. Each term in the first polynomial needs to be multiplied by each term in the second polynomial.

### Step 1: Identify the polynomials
- First Polynomial: \(9m^6 + 12m^5\)
- Second Polynomial: \(m^2 + 2m + 1\)

### Step 2: Expand using a table

We can set up a table as follows:
\[
\begin{array}{c|c|c|c}
\times & m^2 & 2m & 1 \\ \hline
9m^6 & 9m^8 & 18m^7 & 9m^6 \\ \hline
12m^5 & 12m^7 & 24m^6 & 12m^5 \\
\end{array}
\]

### Step 3: Combine the terms from the table

Now we add the products from the table:
1. From \(9m^6\):
- \(9m^8 + 18m^7 + 9m^6\)
2. From \(12m^5\):
- \(12m^7 + 24m^6 + 12m^5\)

Now we combine like terms:
- For \(m^8\): \(9m^8\)
- For \(m^7\): \(18m^7 + 12m^7 = 30m^7\)
- For \(m^6\): \(9m^6 + 24m^6 = 33m^6\)
- For \(m^5\): \(12m^5\)

### Conclusion
Putting it all together, the full expansion is:
\[
9m^8 + 30m^7 + 33m^6 + 12m^5
\]

### Step 4: Find the missing term in the original expression
The original expression given is:
\[
-9m^8 - 6m^7 + ? + 12m^5
\]

We can compare the complete expansion \(9m^8 + 30m^7 + 33m^6 + 12m^5\) to the original:

- The \(m^8\) term: \(-9m^8\)
- The \(m^7\) term: \(-6m^7\)
- The \(m^6\) term: (we need to find this)
- The \(m^5\) term: \(12m^5\)

To find the missing term which corresponds to \(m^6\):
The complete term from the expansion is \(33m^6\). Since we are looking for the value in the original expression, it would be:

**Missing term:** \(-33m^6\)

Thus the completed expression is:
\[
-9m^8 - 6m^7 - 33m^6 + 12m^5
\]