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
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For \( 6s^4 \) and \( 6s^3 \): \[ 6s^4 \cdot 6s^3 = 36s^{7} \]
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For \( 6s^4 \) and \( -15s^4 \): \[ 6s^4 \cdot (-15s^4) = -90s^{8} \]
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For \( 15s^3 \) and \( 6s^3 \): \[ 15s^3 \cdot 6s^3 = 90s^{6} \]
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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:
\[ \boxed{-90s^8 - 189s^7 + 90s^6} \]