Question

multiply the polynomials (6s^4+15s^3)(6s3−15s^4) by constructing a table

Answer 1

To multiply the polynomials \( (6s^4 + 15s^3)(6s^3 - 15s^4) \) using a table method, we can set up a grid where we multiply each term of the first polynomial by each term of the second polynomial.

### Step 1: Set up the terms

First, let’s identify the terms of each polynomial:

- **First polynomial**: \( 6s^4, 15s^3 \)
- **Second polynomial**: \( 6s^3, -15s^4 \)

### Step 2: Create the table

We'll construct a table where we multiply each term in the first polynomial by each term in the second polynomial.

| | \( 6s^3 \) | \( -15s^4 \) |
|--------|------------------|-------------------|
| \( 6s^4 \) | \( 36s^7 \) | \( -90s^8 \) |
| \( 15s^3 \) | \( 90s^6 \) | \( -225s^7 \) |

### Step 3: Calculate each entry

1. \( 6s^4 \cdot 6s^3 = 36s^{7} \)
2. \( 6s^4 \cdot (-15s^4) = -90s^{8} \)
3. \( 15s^3 \cdot 6s^3 = 90s^{6} \)
4. \( 15s^3 \cdot (-15s^4) = -225s^{7} \)

### Step 4: Combine like terms

Now we will gather the results from the table:

- The term for \( s^8 \): \( -90s^8 \)
- The term for \( s^7 \): \( 36s^7 - 225s^7 = -189s^7 \)
- The term for \( s^6 \): \( 90s^6 \)

Combining these gives us the final result:

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

### Final Result

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

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