Question

multiply the polynomials (6s^4+15s^3)(6s^3-15s^4) by constructing a table

Answer 1

To multiply the polynomials \( (6s^4 + 15s^3) \) and \( (6s^3 - 15s^4) \), we can create a table to organize our calculations. We will multiply each term in the first polynomial by each term in the second polynomial.

### Step 1: Set up the table

We'll construct a table where we list the terms of the first polynomial across the top and the terms of the second polynomial along the side.

| | \( 6s^3 \) | \( -15s^4 \) |
|----------|------------|--------------|
| \( 6s^4 \) | \( 6s^4 \cdot 6s^3 \) | \( 6s^4 \cdot (-15s^4) \) |
| \( 15s^3 \) | \( 15s^3 \cdot 6s^3 \) | \( 15s^3 \cdot (-15s^4) \) |

### Step 2: Fill in the table with the products

1. **For \( 6s^4 \) and \( 6s^3 \)**:
\[
6s^4 \cdot 6s^3 = 36s^{7}
\]

2. **For \( 6s^4 \) and \( -15s^4 \)**:
\[
6s^4 \cdot (-15s^4) = -90s^{8}
\]

3. **For \( 15s^3 \) and \( 6s^3 \)**:
\[
15s^3 \cdot 6s^3 = 90s^{6}
\]

4. **For \( 15s^3 \) and \( -15s^4 \)**:
\[
15s^3 \cdot (-15s^4) = -225s^{7}
\]

### Step 3: Combine the results

Now, we'll combine all the products:

- From \( 6s^4 \cdot 6s^3 \): \( 36s^7 \)
- From \( 6s^4 \cdot -15s^4 \): \( -90s^8 \)
- From \( 15s^3 \cdot 6s^3 \): \( 90s^6 \)
- From \( 15s^3 \cdot -15s^4 \): \( -225s^7 \)

Now we can organize and combine like terms:

\[
-90s^8 + (36s^7 - 225s^7) + 90s^6 = -90s^8 - 189s^7 + 90s^6
\]

### Final Result

Thus, the product of the polynomials \( (6s^4 + 15s^3)(6s^3 - 15s^4) \) is:

\[
\boxed{-90s^8 - 189s^7 + 90s^6}
\]