To multiply the two polynomials \((5y^2 - 6y)(7y^7 - y^3)\), you need to apply the distributive property (also known as the FOIL method for binomials). We'll multiply each term in the first polynomial by each term in the second polynomial, and then combine like terms.
Let's do the multiplication step by step:
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Multiply \(5y^2\) by \(7y^7\): \[ 5y^2 \cdot 7y^7 = 35y^{2 + 7} = 35y^9 \]
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Multiply \(5y^2\) by \(-y^3\): \[ 5y^2 \cdot (-y^3) = -5y^{2 + 3} = -5y^5 \]
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Multiply \(-6y\) by \(7y^7\): \[ -6y \cdot 7y^7 = -42y^{1 + 7} = -42y^8 \]
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Multiply \(-6y\) by \(-y^3\): \[ -6y \cdot (-y^3) = 6y^{1 + 3} = 6y^4 \]
Now, combine all the results:
- From step 1: \(35y^9\)
- From step 2: \(-5y^5\)
- From step 3: \(-42y^8\)
- From step 4: \(6y^4\)
Now we write the combined expression:
\[ 35y^9 - 42y^8 - 5y^5 + 6y^4 \]
Finally, we can present the final result in descending order of exponents:
\[ 35y^9 - 42y^8 - 5y^5 + 6y^4 \]
This is the product of \((5y^2 - 6y)(7y^7 - y^3)\).
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