To multiply the polynomials \((6s^4 + 15s^3)(6s^3 - 15s^4)\) using a table, we can follow these steps:
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Create a multiplication table: List each term in the first polynomial across the top and each term in the second polynomial down the side.
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Multiply: Fill in the table by multiplying each term in the first polynomial by each term in the second polynomial.
Step 1: Set up the table
\[ \begin{array}{c|c|c} & 6s^3 & -15s^4 \ \hline 6s^4 & 36s^7 & -90s^8 \ \hline 15s^3 & 90s^6 & -225s^7 \ \end{array} \]
Step 2: Fill in the products
- For \(6s^4 \cdot 6s^3 = 36s^7\)
- For \(6s^4 \cdot (-15s^4) = -90s^8\)
- For \(15s^3 \cdot 6s^3 = 90s^6\)
- For \(15s^3 \cdot (-15s^4) = -225s^7\)
Step 3: Combine the results
Now we combine these results:
- The terms we have are:
- \(36s^8\)
- \(-90s^8\)
- \(90s^6\)
- \(-225s^7\)
Step 4: Group like terms
Now we group the same powers of \(s\):
- The \(s^8\) terms: \(-90s^8\)
- The \(s^7\) terms: \(36s^7 - 225s^7 = -189s^7\)
- The \(s^6\) terms: \(90s^6\)
Final Result
Putting it all together, we get:
\[ -90s^8 - 189s^7 + 90s^6 \]
So, the answer is:
Response: \(-90s^8 - 189s^7 + 90s^6\).